Merge branch 'mppa-cse2' of gricad-gitlab.univ-grenoble-alpes.fr:sixcy/CompCert into mppa-work

7 files changed, 9599 insertions, 0 deletions

diff --git a/test/monniaux/glpk-4.65/src/npp/npp.h b/test/monniaux/glpk-4.65/src/npp/npp.h
new file mode 100644
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+++ b/test/monniaux/glpk-4.65/src/npp/npp.h
@@ -0,0 +1,645 @@
+/* npp.h (LP/MIP preprocessor) */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#ifndef NPP_H
+#define NPP_H
+
+#include "prob.h"
+
+#if 0 /* 20/XI-2017 */
+typedef struct NPP NPP;
+#else
+typedef struct glp_prep NPP;
+#endif
+typedef struct NPPROW NPPROW;
+typedef struct NPPCOL NPPCOL;
+typedef struct NPPAIJ NPPAIJ;
+typedef struct NPPTSE NPPTSE;
+typedef struct NPPLFE NPPLFE;
+
+#if 0 /* 20/XI-2017 */
+struct NPP
+#else
+struct glp_prep
+#endif
+{     /* LP/MIP preprocessor workspace */
+      /*--------------------------------------------------------------*/
+      /* original problem segment */
+      int orig_dir;
+      /* optimization direction flag:
+         GLP_MIN - minimization
+         GLP_MAX - maximization */
+      int orig_m;
+      /* number of rows */
+      int orig_n;
+      /* number of columns */
+      int orig_nnz;
+      /* number of non-zero constraint coefficients */
+      /*--------------------------------------------------------------*/
+      /* transformed problem segment (always minimization) */
+      DMP *pool;
+      /* memory pool to store problem components */
+      char *name;
+      /* problem name (1 to 255 chars); NULL means no name is assigned
+         to the problem */
+      char *obj;
+      /* objective function name (1 to 255 chars); NULL means no name
+         is assigned to the objective function */
+      double c0;
+      /* constant term of the objective function */
+      int nrows;
+      /* number of rows introduced into the problem; this count
+         increases by one every time a new row is added and never
+         decreases; thus, actual number of rows may be less than nrows
+         due to row deletions */
+      int ncols;
+      /* number of columns introduced into the problem; this count
+         increases by one every time a new column is added and never
+         decreases; thus, actual number of column may be less than
+         ncols due to column deletions */
+      NPPROW *r_head;
+      /* pointer to the beginning of the row list */
+      NPPROW *r_tail;
+      /* pointer to the end of the row list */
+      NPPCOL *c_head;
+      /* pointer to the beginning of the column list */
+      NPPCOL *c_tail;
+      /* pointer to the end of the column list */
+      /*--------------------------------------------------------------*/
+      /* transformation history */
+      DMP *stack;
+      /* memory pool to store transformation entries */
+      NPPTSE *top;
+      /* pointer to most recent transformation entry */
+#if 0 /* 16/XII-2009 */
+      int count[1+25];
+      /* transformation statistics */
+#endif
+      /*--------------------------------------------------------------*/
+      /* resultant (preprocessed) problem segment */
+      int m;
+      /* number of rows */
+      int n;
+      /* number of columns */
+      int nnz;
+      /* number of non-zero constraint coefficients */
+      int *row_ref; /* int row_ref[1+m]; */
+      /* row_ref[i], 1 <= i <= m, is the reference number assigned to
+         a row, which is i-th row of the resultant problem */
+      int *col_ref; /* int col_ref[1+n]; */
+      /* col_ref[j], 1 <= j <= n, is the reference number assigned to
+         a column, which is j-th column of the resultant problem */
+      /*--------------------------------------------------------------*/
+      /* recovered solution segment */
+      int sol;
+      /* solution indicator:
+         GLP_SOL - basic solution
+         GLP_IPT - interior-point solution
+         GLP_MIP - mixed integer solution */
+      int scaling;
+      /* scaling option:
+         GLP_OFF - scaling is disabled
+         GLP_ON  - scaling is enabled */
+      int p_stat;
+      /* status of primal basic solution:
+         GLP_UNDEF  - primal solution is undefined
+         GLP_FEAS   - primal solution is feasible
+         GLP_INFEAS - primal solution is infeasible
+         GLP_NOFEAS - no primal feasible solution exists */
+      int d_stat;
+      /* status of dual basic solution:
+         GLP_UNDEF  - dual solution is undefined
+         GLP_FEAS   - dual solution is feasible
+         GLP_INFEAS - dual solution is infeasible
+         GLP_NOFEAS - no dual feasible solution exists */
+      int t_stat;
+      /* status of interior-point solution:
+         GLP_UNDEF  - interior solution is undefined
+         GLP_OPT    - interior solution is optimal */
+      int i_stat;
+      /* status of mixed integer solution:
+         GLP_UNDEF  - integer solution is undefined
+         GLP_OPT    - integer solution is optimal
+         GLP_FEAS   - integer solution is feasible
+         GLP_NOFEAS - no integer solution exists */
+      char *r_stat; /* char r_stat[1+nrows]; */
+      /* r_stat[i], 1 <= i <= nrows, is status of i-th row:
+         GLP_BS - inactive constraint
+         GLP_NL - active constraint on lower bound
+         GLP_NU - active constraint on upper bound
+         GLP_NF - active free row
+         GLP_NS - active equality constraint */
+      char *c_stat; /* char c_stat[1+nrows]; */
+      /* c_stat[j], 1 <= j <= nrows, is status of j-th column:
+         GLP_BS - basic variable
+         GLP_NL - non-basic variable on lower bound
+         GLP_NU - non-basic variable on upper bound
+         GLP_NF - non-basic free variable
+         GLP_NS - non-basic fixed variable */
+      double *r_pi; /* double r_pi[1+nrows]; */
+      /* r_pi[i], 1 <= i <= nrows, is Lagrange multiplier (dual value)
+         for i-th row (constraint) */
+      double *c_value; /* double c_value[1+ncols]; */
+      /* c_value[j], 1 <= j <= ncols, is primal value of j-th column
+         (structural variable) */
+};
+
+struct NPPROW
+{     /* row (constraint) */
+      int i;
+      /* reference number assigned to the row, 1 <= i <= nrows */
+      char *name;
+      /* row name (1 to 255 chars); NULL means no name is assigned to
+         the row */
+      double lb;
+      /* lower bound; -DBL_MAX means the row has no lower bound */
+      double ub;
+      /* upper bound; +DBL_MAX means the row has no upper bound */
+      NPPAIJ *ptr;
+      /* pointer to the linked list of constraint coefficients */
+      int temp;
+      /* working field used by preprocessor routines */
+      NPPROW *prev;
+      /* pointer to previous row in the row list */
+      NPPROW *next;
+      /* pointer to next row in the row list */
+};
+
+struct NPPCOL
+{     /* column (variable) */
+      int j;
+      /* reference number assigned to the column, 1 <= j <= ncols */
+      char *name;
+      /* column name (1 to 255 chars); NULL means no name is assigned
+         to the column */
+      char is_int;
+      /* 0 means continuous variable; 1 means integer variable */
+      double lb;
+      /* lower bound; -DBL_MAX means the column has no lower bound */
+      double ub;
+      /* upper bound; +DBL_MAX means the column has no upper bound */
+      double coef;
+      /* objective coefficient */
+      NPPAIJ *ptr;
+      /* pointer to the linked list of constraint coefficients */
+      int temp;
+      /* working field used by preprocessor routines */
+#if 1 /* 28/XII-2009 */
+      union
+      {  double ll;
+         /* implied column lower bound */
+         int pos;
+         /* vertex ordinal number corresponding to this binary column
+            in the conflict graph (0, if the vertex does not exist) */
+      }  ll;
+      union
+      {  double uu;
+         /* implied column upper bound */
+         int neg;
+         /* vertex ordinal number corresponding to complement of this
+            binary column in the conflict graph (0, if the vertex does
+            not exist) */
+      }  uu;
+#endif
+      NPPCOL *prev;
+      /* pointer to previous column in the column list */
+      NPPCOL *next;
+      /* pointer to next column in the column list */
+};
+
+struct NPPAIJ
+{     /* constraint coefficient */
+      NPPROW *row;
+      /* pointer to corresponding row */
+      NPPCOL *col;
+      /* pointer to corresponding column */
+      double val;
+      /* (non-zero) coefficient value */
+      NPPAIJ *r_prev;
+      /* pointer to previous coefficient in the same row */
+      NPPAIJ *r_next;
+      /* pointer to next coefficient in the same row */
+      NPPAIJ *c_prev;
+      /* pointer to previous coefficient in the same column */
+      NPPAIJ *c_next;
+      /* pointer to next coefficient in the same column */
+};
+
+struct NPPTSE
+{     /* transformation stack entry */
+      int (*func)(NPP *npp, void *info);
+      /* pointer to routine performing back transformation */
+      void *info;
+      /* pointer to specific info (depends on the transformation) */
+      NPPTSE *link;
+      /* pointer to another entry created *before* this entry */
+};
+
+struct NPPLFE
+{     /* linear form element */
+      int ref;
+      /* row/column reference number */
+      double val;
+      /* (non-zero) coefficient value */
+      NPPLFE *next;
+      /* pointer to another element */
+};
+
+#define npp_create_wksp _glp_npp_create_wksp
+NPP *npp_create_wksp(void);
+/* create LP/MIP preprocessor workspace */
+
+#define npp_insert_row _glp_npp_insert_row
+void npp_insert_row(NPP *npp, NPPROW *row, int where);
+/* insert row to the row list */
+
+#define npp_remove_row _glp_npp_remove_row
+void npp_remove_row(NPP *npp, NPPROW *row);
+/* remove row from the row list */
+
+#define npp_activate_row _glp_npp_activate_row
+void npp_activate_row(NPP *npp, NPPROW *row);
+/* make row active */
+
+#define npp_deactivate_row _glp_npp_deactivate_row
+void npp_deactivate_row(NPP *npp, NPPROW *row);
+/* make row inactive */
+
+#define npp_insert_col _glp_npp_insert_col
+void npp_insert_col(NPP *npp, NPPCOL *col, int where);
+/* insert column to the column list */
+
+#define npp_remove_col _glp_npp_remove_col
+void npp_remove_col(NPP *npp, NPPCOL *col);
+/* remove column from the column list */
+
+#define npp_activate_col _glp_npp_activate_col
+void npp_activate_col(NPP *npp, NPPCOL *col);
+/* make column active */
+
+#define npp_deactivate_col _glp_npp_deactivate_col
+void npp_deactivate_col(NPP *npp, NPPCOL *col);
+/* make column inactive */
+
+#define npp_add_row _glp_npp_add_row
+NPPROW *npp_add_row(NPP *npp);
+/* add new row to the current problem */
+
+#define npp_add_col _glp_npp_add_col
+NPPCOL *npp_add_col(NPP *npp);
+/* add new column to the current problem */
+
+#define npp_add_aij _glp_npp_add_aij
+NPPAIJ *npp_add_aij(NPP *npp, NPPROW *row, NPPCOL *col, double val);
+/* add new element to the constraint matrix */
+
+#define npp_row_nnz _glp_npp_row_nnz
+int npp_row_nnz(NPP *npp, NPPROW *row);
+/* count number of non-zero coefficients in row */
+
+#define npp_col_nnz _glp_npp_col_nnz
+int npp_col_nnz(NPP *npp, NPPCOL *col);
+/* count number of non-zero coefficients in column */
+
+#define npp_push_tse _glp_npp_push_tse
+void *npp_push_tse(NPP *npp, int (*func)(NPP *npp, void *info),
+      int size);
+/* push new entry to the transformation stack */
+
+#define npp_erase_row _glp_npp_erase_row
+void npp_erase_row(NPP *npp, NPPROW *row);
+/* erase row content to make it empty */
+
+#define npp_del_row _glp_npp_del_row
+void npp_del_row(NPP *npp, NPPROW *row);
+/* remove row from the current problem */
+
+#define npp_del_col _glp_npp_del_col
+void npp_del_col(NPP *npp, NPPCOL *col);
+/* remove column from the current problem */
+
+#define npp_del_aij _glp_npp_del_aij
+void npp_del_aij(NPP *npp, NPPAIJ *aij);
+/* remove element from the constraint matrix */
+
+#define npp_load_prob _glp_npp_load_prob
+void npp_load_prob(NPP *npp, glp_prob *orig, int names, int sol,
+      int scaling);
+/* load original problem into the preprocessor workspace */
+
+#define npp_build_prob _glp_npp_build_prob
+void npp_build_prob(NPP *npp, glp_prob *prob);
+/* build resultant (preprocessed) problem */
+
+#define npp_postprocess _glp_npp_postprocess
+void npp_postprocess(NPP *npp, glp_prob *prob);
+/* postprocess solution from the resultant problem */
+
+#define npp_unload_sol _glp_npp_unload_sol
+void npp_unload_sol(NPP *npp, glp_prob *orig);
+/* store solution to the original problem */
+
+#define npp_delete_wksp _glp_npp_delete_wksp
+void npp_delete_wksp(NPP *npp);
+/* delete LP/MIP preprocessor workspace */
+
+#define npp_error()
+
+#define npp_free_row _glp_npp_free_row
+void npp_free_row(NPP *npp, NPPROW *p);
+/* process free (unbounded) row */
+
+#define npp_geq_row _glp_npp_geq_row
+void npp_geq_row(NPP *npp, NPPROW *p);
+/* process row of 'not less than' type */
+
+#define npp_leq_row _glp_npp_leq_row
+void npp_leq_row(NPP *npp, NPPROW *p);
+/* process row of 'not greater than' type */
+
+#define npp_free_col _glp_npp_free_col
+void npp_free_col(NPP *npp, NPPCOL *q);
+/* process free (unbounded) column */
+
+#define npp_lbnd_col _glp_npp_lbnd_col
+void npp_lbnd_col(NPP *npp, NPPCOL *q);
+/* process column with (non-zero) lower bound */
+
+#define npp_ubnd_col _glp_npp_ubnd_col
+void npp_ubnd_col(NPP *npp, NPPCOL *q);
+/* process column with upper bound */
+
+#define npp_dbnd_col _glp_npp_dbnd_col
+void npp_dbnd_col(NPP *npp, NPPCOL *q);
+/* process non-negative column with upper bound */
+
+#define npp_fixed_col _glp_npp_fixed_col
+void npp_fixed_col(NPP *npp, NPPCOL *q);
+/* process fixed column */
+
+#define npp_make_equality _glp_npp_make_equality
+int npp_make_equality(NPP *npp, NPPROW *p);
+/* process row with almost identical bounds */
+
+#define npp_make_fixed _glp_npp_make_fixed
+int npp_make_fixed(NPP *npp, NPPCOL *q);
+/* process column with almost identical bounds */
+
+#define npp_empty_row _glp_npp_empty_row
+int npp_empty_row(NPP *npp, NPPROW *p);
+/* process empty row */
+
+#define npp_empty_col _glp_npp_empty_col
+int npp_empty_col(NPP *npp, NPPCOL *q);
+/* process empty column */
+
+#define npp_implied_value _glp_npp_implied_value
+int npp_implied_value(NPP *npp, NPPCOL *q, double s);
+/* process implied column value */
+
+#define npp_eq_singlet _glp_npp_eq_singlet
+int npp_eq_singlet(NPP *npp, NPPROW *p);
+/* process row singleton (equality constraint) */
+
+#define npp_implied_lower _glp_npp_implied_lower
+int npp_implied_lower(NPP *npp, NPPCOL *q, double l);
+/* process implied column lower bound */
+
+#define npp_implied_upper _glp_npp_implied_upper
+int npp_implied_upper(NPP *npp, NPPCOL *q, double u);
+/* process implied upper bound of column */
+
+#define npp_ineq_singlet _glp_npp_ineq_singlet
+int npp_ineq_singlet(NPP *npp, NPPROW *p);
+/* process row singleton (inequality constraint) */
+
+#define npp_implied_slack _glp_npp_implied_slack
+void npp_implied_slack(NPP *npp, NPPCOL *q);
+/* process column singleton (implied slack variable) */
+
+#define npp_implied_free _glp_npp_implied_free
+int npp_implied_free(NPP *npp, NPPCOL *q);
+/* process column singleton (implied free variable) */
+
+#define npp_eq_doublet _glp_npp_eq_doublet
+NPPCOL *npp_eq_doublet(NPP *npp, NPPROW *p);
+/* process row doubleton (equality constraint) */
+
+#define npp_forcing_row _glp_npp_forcing_row
+int npp_forcing_row(NPP *npp, NPPROW *p, int at);
+/* process forcing row */
+
+#define npp_analyze_row _glp_npp_analyze_row
+int npp_analyze_row(NPP *npp, NPPROW *p);
+/* perform general row analysis */
+
+#define npp_inactive_bound _glp_npp_inactive_bound
+void npp_inactive_bound(NPP *npp, NPPROW *p, int which);
+/* remove row lower/upper inactive bound */
+
+#define npp_implied_bounds _glp_npp_implied_bounds
+void npp_implied_bounds(NPP *npp, NPPROW *p);
+/* determine implied column bounds */
+
+#define npp_binarize_prob _glp_npp_binarize_prob
+int npp_binarize_prob(NPP *npp);
+/* binarize MIP problem */
+
+#define npp_is_packing _glp_npp_is_packing
+int npp_is_packing(NPP *npp, NPPROW *row);
+/* test if constraint is packing inequality */
+
+#define npp_hidden_packing _glp_npp_hidden_packing
+int npp_hidden_packing(NPP *npp, NPPROW *row);
+/* identify hidden packing inequality */
+
+#define npp_implied_packing _glp_npp_implied_packing
+int npp_implied_packing(NPP *npp, NPPROW *row, int which,
+      NPPCOL *var[], char set[]);
+/* identify implied packing inequality */
+
+#define npp_is_covering _glp_npp_is_covering
+int npp_is_covering(NPP *npp, NPPROW *row);
+/* test if constraint is covering inequality */
+
+#define npp_hidden_covering _glp_npp_hidden_covering
+int npp_hidden_covering(NPP *npp, NPPROW *row);
+/* identify hidden covering inequality */
+
+#define npp_is_partitioning _glp_npp_is_partitioning
+int npp_is_partitioning(NPP *npp, NPPROW *row);
+/* test if constraint is partitioning equality */
+
+#define npp_reduce_ineq_coef _glp_npp_reduce_ineq_coef
+int npp_reduce_ineq_coef(NPP *npp, NPPROW *row);
+/* reduce inequality constraint coefficients */
+
+#define npp_clean_prob _glp_npp_clean_prob
+void npp_clean_prob(NPP *npp);
+/* perform initial LP/MIP processing */
+
+#define npp_process_row _glp_npp_process_row
+int npp_process_row(NPP *npp, NPPROW *row, int hard);
+/* perform basic row processing */
+
+#define npp_improve_bounds _glp_npp_improve_bounds
+int npp_improve_bounds(NPP *npp, NPPROW *row, int flag);
+/* improve current column bounds */
+
+#define npp_process_col _glp_npp_process_col
+int npp_process_col(NPP *npp, NPPCOL *col);
+/* perform basic column processing */
+
+#define npp_process_prob _glp_npp_process_prob
+int npp_process_prob(NPP *npp, int hard);
+/* perform basic LP/MIP processing */
+
+#define npp_simplex _glp_npp_simplex
+int npp_simplex(NPP *npp, const glp_smcp *parm);
+/* process LP prior to applying primal/dual simplex method */
+
+#define npp_integer _glp_npp_integer
+int npp_integer(NPP *npp, const glp_iocp *parm);
+/* process MIP prior to applying branch-and-bound method */
+
+/**********************************************************************/
+
+#define npp_sat_free_row _glp_npp_sat_free_row
+void npp_sat_free_row(NPP *npp, NPPROW *p);
+/* process free (unbounded) row */
+
+#define npp_sat_fixed_col _glp_npp_sat_fixed_col
+int npp_sat_fixed_col(NPP *npp, NPPCOL *q);
+/* process fixed column */
+
+#define npp_sat_is_bin_comb _glp_npp_sat_is_bin_comb
+int npp_sat_is_bin_comb(NPP *npp, NPPROW *row);
+/* test if row is binary combination */
+
+#define npp_sat_num_pos_coef _glp_npp_sat_num_pos_coef
+int npp_sat_num_pos_coef(NPP *npp, NPPROW *row);
+/* determine number of positive coefficients */
+
+#define npp_sat_num_neg_coef _glp_npp_sat_num_neg_coef
+int npp_sat_num_neg_coef(NPP *npp, NPPROW *row);
+/* determine number of negative coefficients */
+
+#define npp_sat_is_cover_ineq _glp_npp_sat_is_cover_ineq
+int npp_sat_is_cover_ineq(NPP *npp, NPPROW *row);
+/* test if row is covering inequality */
+
+#define npp_sat_is_pack_ineq _glp_npp_sat_is_pack_ineq
+int npp_sat_is_pack_ineq(NPP *npp, NPPROW *row);
+/* test if row is packing inequality */
+
+#define npp_sat_is_partn_eq _glp_npp_sat_is_partn_eq
+int npp_sat_is_partn_eq(NPP *npp, NPPROW *row);
+/* test if row is partitioning equality */
+
+#define npp_sat_reverse_row _glp_npp_sat_reverse_row
+int npp_sat_reverse_row(NPP *npp, NPPROW *row);
+/* multiply both sides of row by -1 */
+
+#define npp_sat_split_pack _glp_npp_sat_split_pack
+NPPROW *npp_sat_split_pack(NPP *npp, NPPROW *row, int nnn);
+/* split packing inequality */
+
+#define npp_sat_encode_pack _glp_npp_sat_encode_pack
+void npp_sat_encode_pack(NPP *npp, NPPROW *row);
+/* encode packing inequality */
+
+typedef struct NPPLIT NPPLIT;
+typedef struct NPPLSE NPPLSE;
+typedef struct NPPSED NPPSED;
+
+struct NPPLIT
+{     /* literal (binary variable or its negation) */
+      NPPCOL *col;
+      /* pointer to binary variable; NULL means constant false */
+      int neg;
+      /* negation flag:
+         0 - literal is variable (or constant false)
+         1 - literal is negation of variable (or constant true) */
+};
+
+struct NPPLSE
+{     /* literal set element */
+      NPPLIT lit;
+      /* literal */
+      NPPLSE *next;
+      /* pointer to another element */
+};
+
+struct NPPSED
+{     /* summation encoding descriptor */
+      /* this struct describes the equality
+            x + y + z = s + 2 * c,
+         which was encoded as CNF and included into the transformed
+         problem; here x and y are literals, z is either a literal or
+         constant zero, s and c are binary variables modeling, resp.,
+         the low and high (carry) sum bits */
+      NPPLIT x, y, z;
+      /* literals; if z.col = NULL, z is constant zero */
+      NPPCOL *s, *c;
+      /* binary variables modeling the sum bits */
+};
+
+#define npp_sat_encode_sum2 _glp_npp_sat_encode_sum2
+void npp_sat_encode_sum2(NPP *npp, NPPLSE *set, NPPSED *sed);
+/* encode 2-bit summation */
+
+#define npp_sat_encode_sum3 _glp_npp_sat_encode_sum3
+void npp_sat_encode_sum3(NPP *npp, NPPLSE *set, NPPSED *sed);
+/* encode 3-bit summation */
+
+#define npp_sat_encode_sum_ax _glp_npp_sat_encode_sum_ax
+int npp_sat_encode_sum_ax(NPP *npp, NPPROW *row, NPPLIT y[]);
+/* encode linear combination of 0-1 variables */
+
+#define npp_sat_normalize_clause _glp_npp_sat_normalize_clause
+int npp_sat_normalize_clause(NPP *npp, int size, NPPLIT lit[]);
+/* normalize clause */
+
+#define npp_sat_encode_clause _glp_npp_sat_encode_clause
+NPPROW *npp_sat_encode_clause(NPP *npp, int size, NPPLIT lit[]);
+/* translate clause to cover inequality */
+
+#define npp_sat_encode_geq _glp_npp_sat_encode_geq
+int npp_sat_encode_geq(NPP *npp, int n, NPPLIT y[], int rhs);
+/* encode "not less than" constraint */
+
+#define npp_sat_encode_leq _glp_npp_sat_encode_leq
+int npp_sat_encode_leq(NPP *npp, int n, NPPLIT y[], int rhs);
+/* encode "not greater than" constraint */
+
+#define npp_sat_encode_row _glp_npp_sat_encode_row
+int npp_sat_encode_row(NPP *npp, NPPROW *row);
+/* encode constraint (row) of general type */
+
+#define npp_sat_encode_prob _glp_npp_sat_encode_prob
+int npp_sat_encode_prob(NPP *npp);
+/* encode 0-1 feasibility problem */
+
+#endif
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp1.c b/test/monniaux/glpk-4.65/src/npp/npp1.c
new file mode 100644
index 00000000..51758bad
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp1.c
@@ -0,0 +1,937 @@
+/* npp1.c */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+NPP *npp_create_wksp(void)
+{     /* create LP/MIP preprocessor workspace */
+      NPP *npp;
+      npp = xmalloc(sizeof(NPP));
+      npp->orig_dir = 0;
+      npp->orig_m = npp->orig_n = npp->orig_nnz = 0;
+      npp->pool = dmp_create_pool();
+      npp->name = npp->obj = NULL;
+      npp->c0 = 0.0;
+      npp->nrows = npp->ncols = 0;
+      npp->r_head = npp->r_tail = NULL;
+      npp->c_head = npp->c_tail = NULL;
+      npp->stack = dmp_create_pool();
+      npp->top = NULL;
+#if 0 /* 16/XII-2009 */
+      memset(&npp->count, 0, sizeof(npp->count));
+#endif
+      npp->m = npp->n = npp->nnz = 0;
+      npp->row_ref = npp->col_ref = NULL;
+      npp->sol = npp->scaling = 0;
+      npp->p_stat = npp->d_stat = npp->t_stat = npp->i_stat = 0;
+      npp->r_stat = NULL;
+      /*npp->r_prim =*/ npp->r_pi = NULL;
+      npp->c_stat = NULL;
+      npp->c_value = /*npp->c_dual =*/ NULL;
+      return npp;
+}
+
+void npp_insert_row(NPP *npp, NPPROW *row, int where)
+{     /* insert row to the row list */
+      if (where == 0)
+      {  /* insert row to the beginning of the row list */
+         row->prev = NULL;
+         row->next = npp->r_head;
+         if (row->next == NULL)
+            npp->r_tail = row;
+         else
+            row->next->prev = row;
+         npp->r_head = row;
+      }
+      else
+      {  /* insert row to the end of the row list */
+         row->prev = npp->r_tail;
+         row->next = NULL;
+         if (row->prev == NULL)
+            npp->r_head = row;
+         else
+            row->prev->next = row;
+         npp->r_tail = row;
+      }
+      return;
+}
+
+void npp_remove_row(NPP *npp, NPPROW *row)
+{     /* remove row from the row list */
+      if (row->prev == NULL)
+         npp->r_head = row->next;
+      else
+         row->prev->next = row->next;
+      if (row->next == NULL)
+         npp->r_tail = row->prev;
+      else
+         row->next->prev = row->prev;
+      return;
+}
+
+void npp_activate_row(NPP *npp, NPPROW *row)
+{     /* make row active */
+      if (!row->temp)
+      {  row->temp = 1;
+         /* move the row to the beginning of the row list */
+         npp_remove_row(npp, row);
+         npp_insert_row(npp, row, 0);
+      }
+      return;
+}
+
+void npp_deactivate_row(NPP *npp, NPPROW *row)
+{     /* make row inactive */
+      if (row->temp)
+      {  row->temp = 0;
+         /* move the row to the end of the row list */
+         npp_remove_row(npp, row);
+         npp_insert_row(npp, row, 1);
+      }
+      return;
+}
+
+void npp_insert_col(NPP *npp, NPPCOL *col, int where)
+{     /* insert column to the column list */
+      if (where == 0)
+      {  /* insert column to the beginning of the column list */
+         col->prev = NULL;
+         col->next = npp->c_head;
+         if (col->next == NULL)
+            npp->c_tail = col;
+         else
+            col->next->prev = col;
+         npp->c_head = col;
+      }
+      else
+      {  /* insert column to the end of the column list */
+         col->prev = npp->c_tail;
+         col->next = NULL;
+         if (col->prev == NULL)
+            npp->c_head = col;
+         else
+            col->prev->next = col;
+         npp->c_tail = col;
+      }
+      return;
+}
+
+void npp_remove_col(NPP *npp, NPPCOL *col)
+{     /* remove column from the column list */
+      if (col->prev == NULL)
+         npp->c_head = col->next;
+      else
+         col->prev->next = col->next;
+      if (col->next == NULL)
+         npp->c_tail = col->prev;
+      else
+         col->next->prev = col->prev;
+      return;
+}
+
+void npp_activate_col(NPP *npp, NPPCOL *col)
+{     /* make column active */
+      if (!col->temp)
+      {  col->temp = 1;
+         /* move the column to the beginning of the column list */
+         npp_remove_col(npp, col);
+         npp_insert_col(npp, col, 0);
+      }
+      return;
+}
+
+void npp_deactivate_col(NPP *npp, NPPCOL *col)
+{     /* make column inactive */
+      if (col->temp)
+      {  col->temp = 0;
+         /* move the column to the end of the column list */
+         npp_remove_col(npp, col);
+         npp_insert_col(npp, col, 1);
+      }
+      return;
+}
+
+NPPROW *npp_add_row(NPP *npp)
+{     /* add new row to the current problem */
+      NPPROW *row;
+      row = dmp_get_atom(npp->pool, sizeof(NPPROW));
+      row->i = ++(npp->nrows);
+      row->name = NULL;
+      row->lb = -DBL_MAX, row->ub = +DBL_MAX;
+      row->ptr = NULL;
+      row->temp = 0;
+      npp_insert_row(npp, row, 1);
+      return row;
+}
+
+NPPCOL *npp_add_col(NPP *npp)
+{     /* add new column to the current problem */
+      NPPCOL *col;
+      col = dmp_get_atom(npp->pool, sizeof(NPPCOL));
+      col->j = ++(npp->ncols);
+      col->name = NULL;
+#if 0
+      col->kind = GLP_CV;
+#else
+      col->is_int = 0;
+#endif
+      col->lb = col->ub = col->coef = 0.0;
+      col->ptr = NULL;
+      col->temp = 0;
+      npp_insert_col(npp, col, 1);
+      return col;
+}
+
+NPPAIJ *npp_add_aij(NPP *npp, NPPROW *row, NPPCOL *col, double val)
+{     /* add new element to the constraint matrix */
+      NPPAIJ *aij;
+      aij = dmp_get_atom(npp->pool, sizeof(NPPAIJ));
+      aij->row = row;
+      aij->col = col;
+      aij->val = val;
+      aij->r_prev = NULL;
+      aij->r_next = row->ptr;
+      aij->c_prev = NULL;
+      aij->c_next = col->ptr;
+      if (aij->r_next != NULL)
+         aij->r_next->r_prev = aij;
+      if (aij->c_next != NULL)
+         aij->c_next->c_prev = aij;
+      row->ptr = col->ptr = aij;
+      return aij;
+}
+
+int npp_row_nnz(NPP *npp, NPPROW *row)
+{     /* count number of non-zero coefficients in row */
+      NPPAIJ *aij;
+      int nnz;
+      xassert(npp == npp);
+      nnz = 0;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+         nnz++;
+      return nnz;
+}
+
+int npp_col_nnz(NPP *npp, NPPCOL *col)
+{     /* count number of non-zero coefficients in column */
+      NPPAIJ *aij;
+      int nnz;
+      xassert(npp == npp);
+      nnz = 0;
+      for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+         nnz++;
+      return nnz;
+}
+
+void *npp_push_tse(NPP *npp, int (*func)(NPP *npp, void *info),
+      int size)
+{     /* push new entry to the transformation stack */
+      NPPTSE *tse;
+      tse = dmp_get_atom(npp->stack, sizeof(NPPTSE));
+      tse->func = func;
+      tse->info = dmp_get_atom(npp->stack, size);
+      tse->link = npp->top;
+      npp->top = tse;
+      return tse->info;
+}
+
+#if 1 /* 23/XII-2009 */
+void npp_erase_row(NPP *npp, NPPROW *row)
+{     /* erase row content to make it empty */
+      NPPAIJ *aij;
+      while (row->ptr != NULL)
+      {  aij = row->ptr;
+         row->ptr = aij->r_next;
+         if (aij->c_prev == NULL)
+            aij->col->ptr = aij->c_next;
+         else
+            aij->c_prev->c_next = aij->c_next;
+         if (aij->c_next == NULL)
+            ;
+         else
+            aij->c_next->c_prev = aij->c_prev;
+         dmp_free_atom(npp->pool, aij, sizeof(NPPAIJ));
+      }
+      return;
+}
+#endif
+
+void npp_del_row(NPP *npp, NPPROW *row)
+{     /* remove row from the current problem */
+#if 0 /* 23/XII-2009 */
+      NPPAIJ *aij;
+#endif
+      if (row->name != NULL)
+         dmp_free_atom(npp->pool, row->name, strlen(row->name)+1);
+#if 0 /* 23/XII-2009 */
+      while (row->ptr != NULL)
+      {  aij = row->ptr;
+         row->ptr = aij->r_next;
+         if (aij->c_prev == NULL)
+            aij->col->ptr = aij->c_next;
+         else
+            aij->c_prev->c_next = aij->c_next;
+         if (aij->c_next == NULL)
+            ;
+         else
+            aij->c_next->c_prev = aij->c_prev;
+         dmp_free_atom(npp->pool, aij, sizeof(NPPAIJ));
+      }
+#else
+      npp_erase_row(npp, row);
+#endif
+      npp_remove_row(npp, row);
+      dmp_free_atom(npp->pool, row, sizeof(NPPROW));
+      return;
+}
+
+void npp_del_col(NPP *npp, NPPCOL *col)
+{     /* remove column from the current problem */
+      NPPAIJ *aij;
+      if (col->name != NULL)
+         dmp_free_atom(npp->pool, col->name, strlen(col->name)+1);
+      while (col->ptr != NULL)
+      {  aij = col->ptr;
+         col->ptr = aij->c_next;
+         if (aij->r_prev == NULL)
+            aij->row->ptr = aij->r_next;
+         else
+            aij->r_prev->r_next = aij->r_next;
+         if (aij->r_next == NULL)
+            ;
+         else
+            aij->r_next->r_prev = aij->r_prev;
+         dmp_free_atom(npp->pool, aij, sizeof(NPPAIJ));
+      }
+      npp_remove_col(npp, col);
+      dmp_free_atom(npp->pool, col, sizeof(NPPCOL));
+      return;
+}
+
+void npp_del_aij(NPP *npp, NPPAIJ *aij)
+{     /* remove element from the constraint matrix */
+      if (aij->r_prev == NULL)
+         aij->row->ptr = aij->r_next;
+      else
+         aij->r_prev->r_next = aij->r_next;
+      if (aij->r_next == NULL)
+         ;
+      else
+         aij->r_next->r_prev = aij->r_prev;
+      if (aij->c_prev == NULL)
+         aij->col->ptr = aij->c_next;
+      else
+         aij->c_prev->c_next = aij->c_next;
+      if (aij->c_next == NULL)
+         ;
+      else
+         aij->c_next->c_prev = aij->c_prev;
+      dmp_free_atom(npp->pool, aij, sizeof(NPPAIJ));
+      return;
+}
+
+void npp_load_prob(NPP *npp, glp_prob *orig, int names, int sol,
+      int scaling)
+{     /* load original problem into the preprocessor workspace */
+      int m = orig->m;
+      int n = orig->n;
+      NPPROW **link;
+      int i, j;
+      double dir;
+      xassert(names == GLP_OFF || names == GLP_ON);
+      xassert(sol == GLP_SOL || sol == GLP_IPT || sol == GLP_MIP);
+      xassert(scaling == GLP_OFF || scaling == GLP_ON);
+      if (sol == GLP_MIP) xassert(!scaling);
+      npp->orig_dir = orig->dir;
+      if (npp->orig_dir == GLP_MIN)
+         dir = +1.0;
+      else if (npp->orig_dir == GLP_MAX)
+         dir = -1.0;
+      else
+         xassert(npp != npp);
+      npp->orig_m = m;
+      npp->orig_n = n;
+      npp->orig_nnz = orig->nnz;
+      if (names && orig->name != NULL)
+      {  npp->name = dmp_get_atom(npp->pool, strlen(orig->name)+1);
+         strcpy(npp->name, orig->name);
+      }
+      if (names && orig->obj != NULL)
+      {  npp->obj = dmp_get_atom(npp->pool, strlen(orig->obj)+1);
+         strcpy(npp->obj, orig->obj);
+      }
+      npp->c0 = dir * orig->c0;
+      /* load rows */
+      link = xcalloc(1+m, sizeof(NPPROW *));
+      for (i = 1; i <= m; i++)
+      {  GLPROW *rrr = orig->row[i];
+         NPPROW *row;
+         link[i] = row = npp_add_row(npp);
+         xassert(row->i == i);
+         if (names && rrr->name != NULL)
+         {  row->name = dmp_get_atom(npp->pool, strlen(rrr->name)+1);
+            strcpy(row->name, rrr->name);
+         }
+         if (!scaling)
+         {  if (rrr->type == GLP_FR)
+               row->lb = -DBL_MAX, row->ub = +DBL_MAX;
+            else if (rrr->type == GLP_LO)
+               row->lb = rrr->lb, row->ub = +DBL_MAX;
+            else if (rrr->type == GLP_UP)
+               row->lb = -DBL_MAX, row->ub = rrr->ub;
+            else if (rrr->type == GLP_DB)
+               row->lb = rrr->lb, row->ub = rrr->ub;
+            else if (rrr->type == GLP_FX)
+               row->lb = row->ub = rrr->lb;
+            else
+               xassert(rrr != rrr);
+         }
+         else
+         {  double rii = rrr->rii;
+            if (rrr->type == GLP_FR)
+               row->lb = -DBL_MAX, row->ub = +DBL_MAX;
+            else if (rrr->type == GLP_LO)
+               row->lb = rrr->lb * rii, row->ub = +DBL_MAX;
+            else if (rrr->type == GLP_UP)
+               row->lb = -DBL_MAX, row->ub = rrr->ub * rii;
+            else if (rrr->type == GLP_DB)
+               row->lb = rrr->lb * rii, row->ub = rrr->ub * rii;
+            else if (rrr->type == GLP_FX)
+               row->lb = row->ub = rrr->lb * rii;
+            else
+               xassert(rrr != rrr);
+         }
+      }
+      /* load columns and constraint coefficients */
+      for (j = 1; j <= n; j++)
+      {  GLPCOL *ccc = orig->col[j];
+         GLPAIJ *aaa;
+         NPPCOL *col;
+         col = npp_add_col(npp);
+         xassert(col->j == j);
+         if (names && ccc->name != NULL)
+         {  col->name = dmp_get_atom(npp->pool, strlen(ccc->name)+1);
+            strcpy(col->name, ccc->name);
+         }
+         if (sol == GLP_MIP)
+#if 0
+            col->kind = ccc->kind;
+#else
+            col->is_int = (char)(ccc->kind == GLP_IV);
+#endif
+         if (!scaling)
+         {  if (ccc->type == GLP_FR)
+               col->lb = -DBL_MAX, col->ub = +DBL_MAX;
+            else if (ccc->type == GLP_LO)
+               col->lb = ccc->lb, col->ub = +DBL_MAX;
+            else if (ccc->type == GLP_UP)
+               col->lb = -DBL_MAX, col->ub = ccc->ub;
+            else if (ccc->type == GLP_DB)
+               col->lb = ccc->lb, col->ub = ccc->ub;
+            else if (ccc->type == GLP_FX)
+               col->lb = col->ub = ccc->lb;
+            else
+               xassert(ccc != ccc);
+            col->coef = dir * ccc->coef;
+            for (aaa = ccc->ptr; aaa != NULL; aaa = aaa->c_next)
+               npp_add_aij(npp, link[aaa->row->i], col, aaa->val);
+         }
+         else
+         {  double sjj = ccc->sjj;
+            if (ccc->type == GLP_FR)
+               col->lb = -DBL_MAX, col->ub = +DBL_MAX;
+            else if (ccc->type == GLP_LO)
+               col->lb = ccc->lb / sjj, col->ub = +DBL_MAX;
+            else if (ccc->type == GLP_UP)
+               col->lb = -DBL_MAX, col->ub = ccc->ub / sjj;
+            else if (ccc->type == GLP_DB)
+               col->lb = ccc->lb / sjj, col->ub = ccc->ub / sjj;
+            else if (ccc->type == GLP_FX)
+               col->lb = col->ub = ccc->lb / sjj;
+            else
+               xassert(ccc != ccc);
+            col->coef = dir * ccc->coef * sjj;
+            for (aaa = ccc->ptr; aaa != NULL; aaa = aaa->c_next)
+               npp_add_aij(npp, link[aaa->row->i], col,
+                  aaa->row->rii * aaa->val * sjj);
+         }
+      }
+      xfree(link);
+      /* keep solution indicator and scaling option */
+      npp->sol = sol;
+      npp->scaling = scaling;
+      return;
+}
+
+void npp_build_prob(NPP *npp, glp_prob *prob)
+{     /* build resultant (preprocessed) problem */
+      NPPROW *row;
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int i, j, type, len, *ind;
+      double dir, *val;
+      glp_erase_prob(prob);
+      glp_set_prob_name(prob, npp->name);
+      glp_set_obj_name(prob, npp->obj);
+      glp_set_obj_dir(prob, npp->orig_dir);
+      if (npp->orig_dir == GLP_MIN)
+         dir = +1.0;
+      else if (npp->orig_dir == GLP_MAX)
+         dir = -1.0;
+      else
+         xassert(npp != npp);
+      glp_set_obj_coef(prob, 0, dir * npp->c0);
+      /* build rows */
+      for (row = npp->r_head; row != NULL; row = row->next)
+      {  row->temp = i = glp_add_rows(prob, 1);
+         glp_set_row_name(prob, i, row->name);
+         if (row->lb == -DBL_MAX && row->ub == +DBL_MAX)
+            type = GLP_FR;
+         else if (row->ub == +DBL_MAX)
+            type = GLP_LO;
+         else if (row->lb == -DBL_MAX)
+            type = GLP_UP;
+         else if (row->lb != row->ub)
+            type = GLP_DB;
+         else
+            type = GLP_FX;
+         glp_set_row_bnds(prob, i, type, row->lb, row->ub);
+      }
+      /* build columns and the constraint matrix */
+      ind = xcalloc(1+prob->m, sizeof(int));
+      val = xcalloc(1+prob->m, sizeof(double));
+      for (col = npp->c_head; col != NULL; col = col->next)
+      {  j = glp_add_cols(prob, 1);
+         glp_set_col_name(prob, j, col->name);
+#if 0
+         glp_set_col_kind(prob, j, col->kind);
+#else
+         glp_set_col_kind(prob, j, col->is_int ? GLP_IV : GLP_CV);
+#endif
+         if (col->lb == -DBL_MAX && col->ub == +DBL_MAX)
+            type = GLP_FR;
+         else if (col->ub == +DBL_MAX)
+            type = GLP_LO;
+         else if (col->lb == -DBL_MAX)
+            type = GLP_UP;
+         else if (col->lb != col->ub)
+            type = GLP_DB;
+         else
+            type = GLP_FX;
+         glp_set_col_bnds(prob, j, type, col->lb, col->ub);
+         glp_set_obj_coef(prob, j, dir * col->coef);
+         len = 0;
+         for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+         {  len++;
+            ind[len] = aij->row->temp;
+            val[len] = aij->val;
+         }
+         glp_set_mat_col(prob, j, len, ind, val);
+      }
+      xfree(ind);
+      xfree(val);
+      /* resultant problem has been built */
+      npp->m = prob->m;
+      npp->n = prob->n;
+      npp->nnz = prob->nnz;
+      npp->row_ref = xcalloc(1+npp->m, sizeof(int));
+      npp->col_ref = xcalloc(1+npp->n, sizeof(int));
+      for (row = npp->r_head, i = 0; row != NULL; row = row->next)
+         npp->row_ref[++i] = row->i;
+      for (col = npp->c_head, j = 0; col != NULL; col = col->next)
+         npp->col_ref[++j] = col->j;
+      /* transformed problem segment is no longer needed */
+      dmp_delete_pool(npp->pool), npp->pool = NULL;
+      npp->name = npp->obj = NULL;
+      npp->c0 = 0.0;
+      npp->r_head = npp->r_tail = NULL;
+      npp->c_head = npp->c_tail = NULL;
+      return;
+}
+
+void npp_postprocess(NPP *npp, glp_prob *prob)
+{     /* postprocess solution from the resultant problem */
+      GLPROW *row;
+      GLPCOL *col;
+      NPPTSE *tse;
+      int i, j, k;
+      double dir;
+      xassert(npp->orig_dir == prob->dir);
+      if (npp->orig_dir == GLP_MIN)
+         dir = +1.0;
+      else if (npp->orig_dir == GLP_MAX)
+         dir = -1.0;
+      else
+         xassert(npp != npp);
+#if 0 /* 11/VII-2013; due to call from ios_main */
+      xassert(npp->m == prob->m);
+#else
+      if (npp->sol != GLP_MIP)
+         xassert(npp->m == prob->m);
+#endif
+      xassert(npp->n == prob->n);
+#if 0 /* 11/VII-2013; due to call from ios_main */
+      xassert(npp->nnz == prob->nnz);
+#else
+      if (npp->sol != GLP_MIP)
+         xassert(npp->nnz == prob->nnz);
+#endif
+      /* copy solution status */
+      if (npp->sol == GLP_SOL)
+      {  npp->p_stat = prob->pbs_stat;
+         npp->d_stat = prob->dbs_stat;
+      }
+      else if (npp->sol == GLP_IPT)
+         npp->t_stat = prob->ipt_stat;
+      else if (npp->sol == GLP_MIP)
+         npp->i_stat = prob->mip_stat;
+      else
+         xassert(npp != npp);
+      /* allocate solution arrays */
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat == NULL)
+            npp->r_stat = xcalloc(1+npp->nrows, sizeof(char));
+         for (i = 1; i <= npp->nrows; i++)
+            npp->r_stat[i] = 0;
+         if (npp->c_stat == NULL)
+            npp->c_stat = xcalloc(1+npp->ncols, sizeof(char));
+         for (j = 1; j <= npp->ncols; j++)
+            npp->c_stat[j] = 0;
+      }
+#if 0
+      if (npp->r_prim == NULL)
+         npp->r_prim = xcalloc(1+npp->nrows, sizeof(double));
+      for (i = 1; i <= npp->nrows; i++)
+         npp->r_prim[i] = DBL_MAX;
+#endif
+      if (npp->c_value == NULL)
+         npp->c_value = xcalloc(1+npp->ncols, sizeof(double));
+      for (j = 1; j <= npp->ncols; j++)
+         npp->c_value[j] = DBL_MAX;
+      if (npp->sol != GLP_MIP)
+      {  if (npp->r_pi == NULL)
+            npp->r_pi = xcalloc(1+npp->nrows, sizeof(double));
+         for (i = 1; i <= npp->nrows; i++)
+            npp->r_pi[i] = DBL_MAX;
+#if 0
+         if (npp->c_dual == NULL)
+            npp->c_dual = xcalloc(1+npp->ncols, sizeof(double));
+         for (j = 1; j <= npp->ncols; j++)
+            npp->c_dual[j] = DBL_MAX;
+#endif
+      }
+      /* copy solution components from the resultant problem */
+      if (npp->sol == GLP_SOL)
+      {  for (i = 1; i <= npp->m; i++)
+         {  row = prob->row[i];
+            k = npp->row_ref[i];
+            npp->r_stat[k] = (char)row->stat;
+            /*npp->r_prim[k] = row->prim;*/
+            npp->r_pi[k] = dir * row->dual;
+         }
+         for (j = 1; j <= npp->n; j++)
+         {  col = prob->col[j];
+            k = npp->col_ref[j];
+            npp->c_stat[k] = (char)col->stat;
+            npp->c_value[k] = col->prim;
+            /*npp->c_dual[k] = dir * col->dual;*/
+         }
+      }
+      else if (npp->sol == GLP_IPT)
+      {  for (i = 1; i <= npp->m; i++)
+         {  row = prob->row[i];
+            k = npp->row_ref[i];
+            /*npp->r_prim[k] = row->pval;*/
+            npp->r_pi[k] = dir * row->dval;
+         }
+         for (j = 1; j <= npp->n; j++)
+         {  col = prob->col[j];
+            k = npp->col_ref[j];
+            npp->c_value[k] = col->pval;
+            /*npp->c_dual[k] = dir * col->dval;*/
+         }
+      }
+      else if (npp->sol == GLP_MIP)
+      {
+#if 0
+         for (i = 1; i <= npp->m; i++)
+         {  row = prob->row[i];
+            k = npp->row_ref[i];
+            /*npp->r_prim[k] = row->mipx;*/
+         }
+#endif
+         for (j = 1; j <= npp->n; j++)
+         {  col = prob->col[j];
+            k = npp->col_ref[j];
+            npp->c_value[k] = col->mipx;
+         }
+      }
+      else
+         xassert(npp != npp);
+      /* perform postprocessing to construct solution to the original
+         problem */
+      for (tse = npp->top; tse != NULL; tse = tse->link)
+      {  xassert(tse->func != NULL);
+         xassert(tse->func(npp, tse->info) == 0);
+      }
+      return;
+}
+
+void npp_unload_sol(NPP *npp, glp_prob *orig)
+{     /* store solution to the original problem */
+      GLPROW *row;
+      GLPCOL *col;
+      int i, j;
+      double dir;
+      xassert(npp->orig_dir == orig->dir);
+      if (npp->orig_dir == GLP_MIN)
+         dir = +1.0;
+      else if (npp->orig_dir == GLP_MAX)
+         dir = -1.0;
+      else
+         xassert(npp != npp);
+      xassert(npp->orig_m == orig->m);
+      xassert(npp->orig_n == orig->n);
+      xassert(npp->orig_nnz == orig->nnz);
+      if (npp->sol == GLP_SOL)
+      {  /* store basic solution */
+         orig->valid = 0;
+         orig->pbs_stat = npp->p_stat;
+         orig->dbs_stat = npp->d_stat;
+         orig->obj_val = orig->c0;
+         orig->some = 0;
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            row->stat = npp->r_stat[i];
+            if (!npp->scaling)
+            {  /*row->prim = npp->r_prim[i];*/
+               row->dual = dir * npp->r_pi[i];
+            }
+            else
+            {  /*row->prim = npp->r_prim[i] / row->rii;*/
+               row->dual = dir * npp->r_pi[i] * row->rii;
+            }
+            if (row->stat == GLP_BS)
+               row->dual = 0.0;
+            else if (row->stat == GLP_NL)
+            {  xassert(row->type == GLP_LO || row->type == GLP_DB);
+               row->prim = row->lb;
+            }
+            else if (row->stat == GLP_NU)
+            {  xassert(row->type == GLP_UP || row->type == GLP_DB);
+               row->prim = row->ub;
+            }
+            else if (row->stat == GLP_NF)
+            {  xassert(row->type == GLP_FR);
+               row->prim = 0.0;
+            }
+            else if (row->stat == GLP_NS)
+            {  xassert(row->type == GLP_FX);
+               row->prim = row->lb;
+            }
+            else
+               xassert(row != row);
+         }
+         for (j = 1; j <= orig->n; j++)
+         {  col = orig->col[j];
+            col->stat = npp->c_stat[j];
+            if (!npp->scaling)
+            {  col->prim = npp->c_value[j];
+               /*col->dual = dir * npp->c_dual[j];*/
+            }
+            else
+            {  col->prim = npp->c_value[j] * col->sjj;
+               /*col->dual = dir * npp->c_dual[j] / col->sjj;*/
+            }
+            if (col->stat == GLP_BS)
+               col->dual = 0.0;
+#if 1
+            else if (col->stat == GLP_NL)
+            {  xassert(col->type == GLP_LO || col->type == GLP_DB);
+               col->prim = col->lb;
+            }
+            else if (col->stat == GLP_NU)
+            {  xassert(col->type == GLP_UP || col->type == GLP_DB);
+               col->prim = col->ub;
+            }
+            else if (col->stat == GLP_NF)
+            {  xassert(col->type == GLP_FR);
+               col->prim = 0.0;
+            }
+            else if (col->stat == GLP_NS)
+            {  xassert(col->type == GLP_FX);
+               col->prim = col->lb;
+            }
+            else
+               xassert(col != col);
+#endif
+            orig->obj_val += col->coef * col->prim;
+         }
+#if 1
+         /* compute primal values of inactive rows */
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            if (row->stat == GLP_BS)
+            {  GLPAIJ *aij;
+               double temp;
+               temp = 0.0;
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  temp += aij->val * aij->col->prim;
+               row->prim = temp;
+            }
+         }
+         /* compute reduced costs of active columns */
+         for (j = 1; j <= orig->n; j++)
+         {  col = orig->col[j];
+            if (col->stat != GLP_BS)
+            {  GLPAIJ *aij;
+               double temp;
+               temp = col->coef;
+               for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+                  temp -= aij->val * aij->row->dual;
+               col->dual = temp;
+            }
+         }
+#endif
+      }
+      else if (npp->sol == GLP_IPT)
+      {  /* store interior-point solution */
+         orig->ipt_stat = npp->t_stat;
+         orig->ipt_obj = orig->c0;
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            if (!npp->scaling)
+            {  /*row->pval = npp->r_prim[i];*/
+               row->dval = dir * npp->r_pi[i];
+            }
+            else
+            {  /*row->pval = npp->r_prim[i] / row->rii;*/
+               row->dval = dir * npp->r_pi[i] * row->rii;
+            }
+         }
+         for (j = 1; j <= orig->n; j++)
+         {  col = orig->col[j];
+            if (!npp->scaling)
+            {  col->pval = npp->c_value[j];
+               /*col->dval = dir * npp->c_dual[j];*/
+            }
+            else
+            {  col->pval = npp->c_value[j] * col->sjj;
+               /*col->dval = dir * npp->c_dual[j] / col->sjj;*/
+            }
+            orig->ipt_obj += col->coef * col->pval;
+         }
+#if 1
+         /* compute row primal values */
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            {  GLPAIJ *aij;
+               double temp;
+               temp = 0.0;
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  temp += aij->val * aij->col->pval;
+               row->pval = temp;
+            }
+         }
+         /* compute column dual values */
+         for (j = 1; j <= orig->n; j++)
+         {  col = orig->col[j];
+            {  GLPAIJ *aij;
+               double temp;
+               temp = col->coef;
+               for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+                  temp -= aij->val * aij->row->dval;
+               col->dval = temp;
+            }
+         }
+#endif
+      }
+      else if (npp->sol == GLP_MIP)
+      {  /* store MIP solution */
+         xassert(!npp->scaling);
+         orig->mip_stat = npp->i_stat;
+         orig->mip_obj = orig->c0;
+#if 0
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            /*row->mipx = npp->r_prim[i];*/
+         }
+#endif
+         for (j = 1; j <= orig->n; j++)
+         {  col = orig->col[j];
+            col->mipx = npp->c_value[j];
+            if (col->kind == GLP_IV)
+               xassert(col->mipx == floor(col->mipx));
+            orig->mip_obj += col->coef * col->mipx;
+         }
+#if 1
+         /* compute row primal values */
+         for (i = 1; i <= orig->m; i++)
+         {  row = orig->row[i];
+            {  GLPAIJ *aij;
+               double temp;
+               temp = 0.0;
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  temp += aij->val * aij->col->mipx;
+               row->mipx = temp;
+            }
+         }
+#endif
+      }
+      else
+         xassert(npp != npp);
+      return;
+}
+
+void npp_delete_wksp(NPP *npp)
+{     /* delete LP/MIP preprocessor workspace */
+      if (npp->pool != NULL)
+         dmp_delete_pool(npp->pool);
+      if (npp->stack != NULL)
+         dmp_delete_pool(npp->stack);
+      if (npp->row_ref != NULL)
+         xfree(npp->row_ref);
+      if (npp->col_ref != NULL)
+         xfree(npp->col_ref);
+      if (npp->r_stat != NULL)
+         xfree(npp->r_stat);
+#if 0
+      if (npp->r_prim != NULL)
+         xfree(npp->r_prim);
+#endif
+      if (npp->r_pi != NULL)
+         xfree(npp->r_pi);
+      if (npp->c_stat != NULL)
+         xfree(npp->c_stat);
+      if (npp->c_value != NULL)
+         xfree(npp->c_value);
+#if 0
+      if (npp->c_dual != NULL)
+         xfree(npp->c_dual);
+#endif
+      xfree(npp);
+      return;
+}
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp2.c b/test/monniaux/glpk-4.65/src/npp/npp2.c
new file mode 100644
index 00000000..4efcf1d1
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp2.c
@@ -0,0 +1,1433 @@
+/* npp2.c */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+/***********************************************************************
+*  NAME
+*
+*  npp_free_row - process free (unbounded) row
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_free_row(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_free_row processes row p, which is free (i.e. has
+*  no finite bounds):
+*
+*     -inf < sum a[p,j] x[j] < +inf.                                 (1)
+*             j
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Constraint (1) cannot be active, so it is redundant and can be
+*  removed from the original problem.
+*
+*  Removing row p leads to removing a column of multiplier pi[p] for
+*  this row in the dual system. Since row p has no bounds, pi[p] = 0,
+*  so removing the column does not affect the dual solution.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  In solution to the original problem row p is inactive constraint,
+*  so it is assigned status GLP_BS, and multiplier pi[p] is assigned
+*  zero value.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  In solution to the original problem row p is inactive constraint,
+*  so its multiplier pi[p] is assigned zero value.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct free_row
+{     /* free (unbounded) row */
+      int p;
+      /* row reference number */
+};
+
+static int rcv_free_row(NPP *npp, void *info);
+
+void npp_free_row(NPP *npp, NPPROW *p)
+{     /* process free (unbounded) row */
+      struct free_row *info;
+      /* the row must be free */
+      xassert(p->lb == -DBL_MAX && p->ub == +DBL_MAX);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_free_row, sizeof(struct free_row));
+      info->p = p->i;
+      /* remove the row from the problem */
+      npp_del_row(npp, p);
+      return;
+}
+
+static int rcv_free_row(NPP *npp, void *_info)
+{     /* recover free (unbounded) row */
+      struct free_row *info = _info;
+      if (npp->sol == GLP_SOL)
+         npp->r_stat[info->p] = GLP_BS;
+      if (npp->sol != GLP_MIP)
+         npp->r_pi[info->p] = 0.0;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_geq_row - process row of 'not less than' type
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_geq_row(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_geq_row processes row p, which is 'not less than'
+*  inequality constraint:
+*
+*     L[p] <= sum a[p,j] x[j] (<= U[p]),                             (1)
+*              j
+*
+*  where L[p] < U[p], and upper bound may not exist (U[p] = +oo).
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Constraint (1) can be replaced by equality constraint:
+*
+*     sum a[p,j] x[j] - s = L[p],                                    (2)
+*      j
+*
+*  where
+*
+*     0 <= s (<= U[p] - L[p])                                        (3)
+*
+*  is a non-negative surplus variable.
+*
+*  Since in the primal system there appears column s having the only
+*  non-zero coefficient in row p, in the dual system there appears a
+*  new row:
+*
+*     (-1) pi[p] + lambda = 0,                                       (4)
+*
+*  where (-1) is coefficient of column s in row p, pi[p] is multiplier
+*  of row p, lambda is multiplier of column q, 0 is coefficient of
+*  column s in the objective row.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of row p in solution to the original problem is determined
+*  by its status and status of column q in solution to the transformed
+*  problem as follows:
+*
+*     +--------------------------------------+------------------+
+*     |         Transformed problem          | Original problem |
+*     +-----------------+--------------------+------------------+
+*     | Status of row p | Status of column s | Status of row p  |
+*     +-----------------+--------------------+------------------+
+*     |     GLP_BS      |       GLP_BS       |       N/A        |
+*     |     GLP_BS      |       GLP_NL       |      GLP_BS      |
+*     |     GLP_BS      |       GLP_NU       |      GLP_BS      |
+*     |     GLP_NS      |       GLP_BS       |      GLP_BS      |
+*     |     GLP_NS      |       GLP_NL       |      GLP_NL      |
+*     |     GLP_NS      |       GLP_NU       |      GLP_NU      |
+*     +-----------------+--------------------+------------------+
+*
+*  Value of row multiplier pi[p] in solution to the original problem
+*  is the same as in solution to the transformed problem.
+*
+*  1. In solution to the transformed problem row p and column q cannot
+*     be basic at the same time; otherwise the basis matrix would have
+*     two linear dependent columns: unity column of auxiliary variable
+*     of row p and unity column of variable s.
+*
+*  2. Though in the transformed problem row p is equality constraint,
+*     it may be basic due to primal degenerate solution.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of row multiplier pi[p] in solution to the original problem
+*  is the same as in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct ineq_row
+{     /* inequality constraint row */
+      int p;
+      /* row reference number */
+      int s;
+      /* column reference number for slack/surplus variable */
+};
+
+static int rcv_geq_row(NPP *npp, void *info);
+
+void npp_geq_row(NPP *npp, NPPROW *p)
+{     /* process row of 'not less than' type */
+      struct ineq_row *info;
+      NPPCOL *s;
+      /* the row must have lower bound */
+      xassert(p->lb != -DBL_MAX);
+      xassert(p->lb < p->ub);
+      /* create column for surplus variable */
+      s = npp_add_col(npp);
+      s->lb = 0.0;
+      s->ub = (p->ub == +DBL_MAX ? +DBL_MAX : p->ub - p->lb);
+      /* and add it to the transformed problem */
+      npp_add_aij(npp, p, s, -1.0);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_geq_row, sizeof(struct ineq_row));
+      info->p = p->i;
+      info->s = s->j;
+      /* replace the row by equality constraint */
+      p->ub = p->lb;
+      return;
+}
+
+static int rcv_geq_row(NPP *npp, void *_info)
+{     /* recover row of 'not less than' type */
+      struct ineq_row *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] == GLP_BS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+            {  npp_error();
+               return 1;
+            }
+            else if (npp->c_stat[info->s] == GLP_NL ||
+                     npp->c_stat[info->s] == GLP_NU)
+               npp->r_stat[info->p] = GLP_BS;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else if (npp->r_stat[info->p] == GLP_NS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+               npp->r_stat[info->p] = GLP_BS;
+            else if (npp->c_stat[info->s] == GLP_NL)
+               npp->r_stat[info->p] = GLP_NL;
+            else if (npp->c_stat[info->s] == GLP_NU)
+               npp->r_stat[info->p] = GLP_NU;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_leq_row - process row of 'not greater than' type
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_leq_row(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_leq_row processes row p, which is 'not greater than'
+*  inequality constraint:
+*
+*     (L[p] <=) sum a[p,j] x[j] <= U[p],                             (1)
+*                j
+*
+*  where L[p] < U[p], and lower bound may not exist (L[p] = +oo).
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Constraint (1) can be replaced by equality constraint:
+*
+*     sum a[p,j] x[j] + s = L[p],                                    (2)
+*      j
+*
+*  where
+*
+*     0 <= s (<= U[p] - L[p])                                        (3)
+*
+*  is a non-negative slack variable.
+*
+*  Since in the primal system there appears column s having the only
+*  non-zero coefficient in row p, in the dual system there appears a
+*  new row:
+*
+*     (+1) pi[p] + lambda = 0,                                       (4)
+*
+*  where (+1) is coefficient of column s in row p, pi[p] is multiplier
+*  of row p, lambda is multiplier of column q, 0 is coefficient of
+*  column s in the objective row.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of row p in solution to the original problem is determined
+*  by its status and status of column q in solution to the transformed
+*  problem as follows:
+*
+*     +--------------------------------------+------------------+
+*     |         Transformed problem          | Original problem |
+*     +-----------------+--------------------+------------------+
+*     | Status of row p | Status of column s | Status of row p  |
+*     +-----------------+--------------------+------------------+
+*     |     GLP_BS      |       GLP_BS       |       N/A        |
+*     |     GLP_BS      |       GLP_NL       |      GLP_BS      |
+*     |     GLP_BS      |       GLP_NU       |      GLP_BS      |
+*     |     GLP_NS      |       GLP_BS       |      GLP_BS      |
+*     |     GLP_NS      |       GLP_NL       |      GLP_NU      |
+*     |     GLP_NS      |       GLP_NU       |      GLP_NL      |
+*     +-----------------+--------------------+------------------+
+*
+*  Value of row multiplier pi[p] in solution to the original problem
+*  is the same as in solution to the transformed problem.
+*
+*  1. In solution to the transformed problem row p and column q cannot
+*     be basic at the same time; otherwise the basis matrix would have
+*     two linear dependent columns: unity column of auxiliary variable
+*     of row p and unity column of variable s.
+*
+*  2. Though in the transformed problem row p is equality constraint,
+*     it may be basic due to primal degeneracy.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of row multiplier pi[p] in solution to the original problem
+*  is the same as in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+static int rcv_leq_row(NPP *npp, void *info);
+
+void npp_leq_row(NPP *npp, NPPROW *p)
+{     /* process row of 'not greater than' type */
+      struct ineq_row *info;
+      NPPCOL *s;
+      /* the row must have upper bound */
+      xassert(p->ub != +DBL_MAX);
+      xassert(p->lb < p->ub);
+      /* create column for slack variable */
+      s = npp_add_col(npp);
+      s->lb = 0.0;
+      s->ub = (p->lb == -DBL_MAX ? +DBL_MAX : p->ub - p->lb);
+      /* and add it to the transformed problem */
+      npp_add_aij(npp, p, s, +1.0);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_leq_row, sizeof(struct ineq_row));
+      info->p = p->i;
+      info->s = s->j;
+      /* replace the row by equality constraint */
+      p->lb = p->ub;
+      return;
+}
+
+static int rcv_leq_row(NPP *npp, void *_info)
+{     /* recover row of 'not greater than' type */
+      struct ineq_row *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] == GLP_BS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+            {  npp_error();
+               return 1;
+            }
+            else if (npp->c_stat[info->s] == GLP_NL ||
+                     npp->c_stat[info->s] == GLP_NU)
+               npp->r_stat[info->p] = GLP_BS;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else if (npp->r_stat[info->p] == GLP_NS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+               npp->r_stat[info->p] = GLP_BS;
+            else if (npp->c_stat[info->s] == GLP_NL)
+               npp->r_stat[info->p] = GLP_NU;
+            else if (npp->c_stat[info->s] == GLP_NU)
+               npp->r_stat[info->p] = GLP_NL;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_free_col - process free (unbounded) column
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_free_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_free_col processes column q, which is free (i.e. has
+*  no finite bounds):
+*
+*     -oo < x[q] < +oo.                                              (1)
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Free (unbounded) variable can be replaced by the difference of two
+*  non-negative variables:
+*
+*     x[q] = s' - s'',   s', s'' >= 0.                               (2)
+*
+*  Assuming that in the transformed problem x[q] becomes s',
+*  transformation (2) causes new column s'' to appear, which differs
+*  from column s' only in the sign of coefficients in constraint and
+*  objective rows. Thus, if in the dual system the following row
+*  corresponds to column s':
+*
+*     sum a[i,q] pi[i] + lambda' = c[q],                             (3)
+*      i
+*
+*  the row which corresponds to column s'' is the following:
+*
+*     sum (-a[i,q]) pi[i] + lambda'' = -c[q].                        (4)
+*      i
+*
+*  Then from (3) and (4) it follows that:
+*
+*     lambda' + lambda'' = 0   =>   lambda' = lmabda'' = 0,          (5)
+*
+*  where lambda' and lambda'' are multipliers for columns s' and s'',
+*  resp.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  With respect to (5) status of column q in solution to the original
+*  problem is determined by statuses of columns s' and s'' in solution
+*  to the transformed problem as follows:
+*
+*     +--------------------------------------+------------------+
+*     |         Transformed problem          | Original problem |
+*     +------------------+-------------------+------------------+
+*     | Status of col s' | Status of col s'' | Status of col q  |
+*     +------------------+-------------------+------------------+
+*     |      GLP_BS      |      GLP_BS       |       N/A        |
+*     |      GLP_BS      |      GLP_NL       |      GLP_BS      |
+*     |      GLP_NL      |      GLP_BS       |      GLP_BS      |
+*     |      GLP_NL      |      GLP_NL       |      GLP_NF      |
+*     +------------------+-------------------+------------------+
+*
+*  Value of column q is computed with formula (2).
+*
+*  1. In solution to the transformed problem columns s' and s'' cannot
+*     be basic at the same time, because they differ only in the sign,
+*     hence, are linear dependent.
+*
+*  2. Though column q is free, it can be non-basic due to dual
+*     degeneracy.
+*
+*  3. If column q is integral, columns s' and s'' are also integral.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q is computed with formula (2).
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is computed with formula (2). */
+
+struct free_col
+{     /* free (unbounded) column */
+      int q;
+      /* column reference number for variables x[q] and s' */
+      int s;
+      /* column reference number for variable s'' */
+};
+
+static int rcv_free_col(NPP *npp, void *info);
+
+void npp_free_col(NPP *npp, NPPCOL *q)
+{     /* process free (unbounded) column */
+      struct free_col *info;
+      NPPCOL *s;
+      NPPAIJ *aij;
+      /* the column must be free */
+      xassert(q->lb == -DBL_MAX && q->ub == +DBL_MAX);
+      /* variable x[q] becomes s' */
+      q->lb = 0.0, q->ub = +DBL_MAX;
+      /* create variable s'' */
+      s = npp_add_col(npp);
+      s->is_int = q->is_int;
+      s->lb = 0.0, s->ub = +DBL_MAX;
+      /* duplicate objective coefficient */
+      s->coef = -q->coef;
+      /* duplicate column of the constraint matrix */
+      for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+         npp_add_aij(npp, aij->row, s, -aij->val);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_free_col, sizeof(struct free_col));
+      info->q = q->j;
+      info->s = s->j;
+      return;
+}
+
+static int rcv_free_col(NPP *npp, void *_info)
+{     /* recover free (unbounded) column */
+      struct free_col *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->c_stat[info->q] == GLP_BS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+            {  npp_error();
+               return 1;
+            }
+            else if (npp->c_stat[info->s] == GLP_NL)
+               npp->c_stat[info->q] = GLP_BS;
+            else
+            {  npp_error();
+               return -1;
+            }
+         }
+         else if (npp->c_stat[info->q] == GLP_NL)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+               npp->c_stat[info->q] = GLP_BS;
+            else if (npp->c_stat[info->s] == GLP_NL)
+               npp->c_stat[info->q] = GLP_NF;
+            else
+            {  npp_error();
+               return -1;
+            }
+         }
+         else
+         {  npp_error();
+            return -1;
+         }
+      }
+      /* compute value of x[q] with formula (2) */
+      npp->c_value[info->q] -= npp->c_value[info->s];
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_lbnd_col - process column with (non-zero) lower bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_lbnd_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_lbnd_col processes column q, which has (non-zero)
+*  lower bound:
+*
+*     l[q] <= x[q] (<= u[q]),                                        (1)
+*
+*  where l[q] < u[q], and upper bound may not exist (u[q] = +oo).
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Column q can be replaced as follows:
+*
+*     x[q] = l[q] + s,                                               (2)
+*
+*  where
+*
+*     0 <= s (<= u[q] - l[q])                                        (3)
+*
+*  is a non-negative variable.
+*
+*  Substituting x[q] from (2) into the objective row, we have:
+*
+*     z = sum c[j] x[j] + c0 =
+*          j
+*
+*       = sum c[j] x[j] + c[q] x[q] + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] + c[q] (l[q] + s) + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] + c[q] s + c~0,
+*
+*  where
+*
+*     c~0 = c0 + c[q] l[q]                                           (4)
+*
+*  is the constant term of the objective in the transformed problem.
+*  Similarly, substituting x[q] into constraint row i, we have:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i]  ==>
+*              j
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] x[q] <= U[i]  ==>
+*             j!=q
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] (l[q] + s) <= U[i]  ==>
+*             j!=q
+*
+*     L~[i] <= sum a[i,j] x[j] + a[i,q] s <= U~[i],
+*              j!=q
+*
+*  where
+*
+*     L~[i] = L[i] - a[i,q] l[q],  U~[i] = U[i] - a[i,q] l[q]        (5)
+*
+*  are lower and upper bounds of row i in the transformed problem,
+*  resp.
+*
+*  Transformation (2) does not affect the dual system.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of column q in solution to the original problem is the same
+*  as in solution to the transformed problem (GLP_BS, GLP_NL or GLP_NU).
+*  Value of column q is computed with formula (2).
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q is computed with formula (2).
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is computed with formula (2). */
+
+struct bnd_col
+{     /* bounded column */
+      int q;
+      /* column reference number for variables x[q] and s */
+      double bnd;
+      /* lower/upper bound l[q] or u[q] */
+};
+
+static int rcv_lbnd_col(NPP *npp, void *info);
+
+void npp_lbnd_col(NPP *npp, NPPCOL *q)
+{     /* process column with (non-zero) lower bound */
+      struct bnd_col *info;
+      NPPROW *i;
+      NPPAIJ *aij;
+      /* the column must have non-zero lower bound */
+      xassert(q->lb != 0.0);
+      xassert(q->lb != -DBL_MAX);
+      xassert(q->lb < q->ub);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_lbnd_col, sizeof(struct bnd_col));
+      info->q = q->j;
+      info->bnd = q->lb;
+      /* substitute x[q] into objective row */
+      npp->c0 += q->coef * q->lb;
+      /* substitute x[q] into constraint rows */
+      for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+      {  i = aij->row;
+         if (i->lb == i->ub)
+            i->ub = (i->lb -= aij->val * q->lb);
+         else
+         {  if (i->lb != -DBL_MAX)
+               i->lb -= aij->val * q->lb;
+            if (i->ub != +DBL_MAX)
+               i->ub -= aij->val * q->lb;
+         }
+      }
+      /* column x[q] becomes column s */
+      if (q->ub != +DBL_MAX)
+         q->ub -= q->lb;
+      q->lb = 0.0;
+      return;
+}
+
+static int rcv_lbnd_col(NPP *npp, void *_info)
+{     /* recover column with (non-zero) lower bound */
+      struct bnd_col *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->c_stat[info->q] == GLP_BS ||
+             npp->c_stat[info->q] == GLP_NL ||
+             npp->c_stat[info->q] == GLP_NU)
+            npp->c_stat[info->q] = npp->c_stat[info->q];
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      /* compute value of x[q] with formula (2) */
+      npp->c_value[info->q] = info->bnd + npp->c_value[info->q];
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_ubnd_col - process column with upper bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_ubnd_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_ubnd_col processes column q, which has upper bound:
+*
+*     (l[q] <=) x[q] <= u[q],                                        (1)
+*
+*  where l[q] < u[q], and lower bound may not exist (l[q] = -oo).
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Column q can be replaced as follows:
+*
+*     x[q] = u[q] - s,                                               (2)
+*
+*  where
+*
+*     0 <= s (<= u[q] - l[q])                                        (3)
+*
+*  is a non-negative variable.
+*
+*  Substituting x[q] from (2) into the objective row, we have:
+*
+*     z = sum c[j] x[j] + c0 =
+*          j
+*
+*       = sum c[j] x[j] + c[q] x[q] + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] + c[q] (u[q] - s) + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] - c[q] s + c~0,
+*
+*  where
+*
+*     c~0 = c0 + c[q] u[q]                                           (4)
+*
+*  is the constant term of the objective in the transformed problem.
+*  Similarly, substituting x[q] into constraint row i, we have:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i]  ==>
+*              j
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] x[q] <= U[i]  ==>
+*             j!=q
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] (u[q] - s) <= U[i]  ==>
+*             j!=q
+*
+*     L~[i] <= sum a[i,j] x[j] - a[i,q] s <= U~[i],
+*              j!=q
+*
+*  where
+*
+*     L~[i] = L[i] - a[i,q] u[q],  U~[i] = U[i] - a[i,q] u[q]        (5)
+*
+*  are lower and upper bounds of row i in the transformed problem,
+*  resp.
+*
+*  Note that in the transformed problem coefficients c[q] and a[i,q]
+*  change their sign. Thus, the row of the dual system corresponding to
+*  column q:
+*
+*     sum a[i,q] pi[i] + lambda[q] = c[q]                            (6)
+*      i
+*
+*  in the transformed problem becomes the following:
+*
+*     sum (-a[i,q]) pi[i] + lambda[s] = -c[q].                       (7)
+*      i
+*
+*  Therefore:
+*
+*     lambda[q] = - lambda[s],                                       (8)
+*
+*  where lambda[q] is multiplier for column q, lambda[s] is multiplier
+*  for column s.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  With respect to (8) status of column q in solution to the original
+*  problem is determined by status of column s in solution to the
+*  transformed problem as follows:
+*
+*     +-----------------------+--------------------+
+*     |  Status of column s   | Status of column q |
+*     | (transformed problem) | (original problem) |
+*     +-----------------------+--------------------+
+*     |        GLP_BS         |       GLP_BS       |
+*     |        GLP_NL         |       GLP_NU       |
+*     |        GLP_NU         |       GLP_NL       |
+*     +-----------------------+--------------------+
+*
+*  Value of column q is computed with formula (2).
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q is computed with formula (2).
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is computed with formula (2). */
+
+static int rcv_ubnd_col(NPP *npp, void *info);
+
+void npp_ubnd_col(NPP *npp, NPPCOL *q)
+{     /* process column with upper bound */
+      struct bnd_col *info;
+      NPPROW *i;
+      NPPAIJ *aij;
+      /* the column must have upper bound */
+      xassert(q->ub != +DBL_MAX);
+      xassert(q->lb < q->ub);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_ubnd_col, sizeof(struct bnd_col));
+      info->q = q->j;
+      info->bnd = q->ub;
+      /* substitute x[q] into objective row */
+      npp->c0 += q->coef * q->ub;
+      q->coef = -q->coef;
+      /* substitute x[q] into constraint rows */
+      for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+      {  i = aij->row;
+         if (i->lb == i->ub)
+            i->ub = (i->lb -= aij->val * q->ub);
+         else
+         {  if (i->lb != -DBL_MAX)
+               i->lb -= aij->val * q->ub;
+            if (i->ub != +DBL_MAX)
+               i->ub -= aij->val * q->ub;
+         }
+         aij->val = -aij->val;
+      }
+      /* column x[q] becomes column s */
+      if (q->lb != -DBL_MAX)
+         q->ub -= q->lb;
+      else
+         q->ub = +DBL_MAX;
+      q->lb = 0.0;
+      return;
+}
+
+static int rcv_ubnd_col(NPP *npp, void *_info)
+{     /* recover column with upper bound */
+      struct bnd_col *info = _info;
+      if (npp->sol == GLP_BS)
+      {  if (npp->c_stat[info->q] == GLP_BS)
+            npp->c_stat[info->q] = GLP_BS;
+         else if (npp->c_stat[info->q] == GLP_NL)
+            npp->c_stat[info->q] = GLP_NU;
+         else if (npp->c_stat[info->q] == GLP_NU)
+            npp->c_stat[info->q] = GLP_NL;
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      /* compute value of x[q] with formula (2) */
+      npp->c_value[info->q] = info->bnd - npp->c_value[info->q];
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_dbnd_col - process non-negative column with upper bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_dbnd_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_dbnd_col processes column q, which is non-negative
+*  and has upper bound:
+*
+*     0 <= x[q] <= u[q],                                             (1)
+*
+*  where u[q] > 0.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Upper bound of column q can be replaced by the following equality
+*  constraint:
+*
+*     x[q] + s = u[q],                                               (2)
+*
+*  where s >= 0 is a non-negative complement variable.
+*
+*  Since in the primal system along with new row (2) there appears a
+*  new column s having the only non-zero coefficient in this row, in
+*  the dual system there appears a new row:
+*
+*     (+1)pi + lambda[s] = 0,                                        (3)
+*
+*  where (+1) is coefficient at column s in row (2), pi is multiplier
+*  for row (2), lambda[s] is multiplier for column s, 0 is coefficient
+*  at column s in the objective row.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of column q in solution to the original problem is determined
+*  by its status and status of column s in solution to the transformed
+*  problem as follows:
+*
+*     +-----------------------------------+------------------+
+*     |         Transformed problem       | Original problem |
+*     +-----------------+-----------------+------------------+
+*     | Status of col q | Status of col s | Status of col q  |
+*     +-----------------+-----------------+------------------+
+*     |     GLP_BS      |     GLP_BS      |      GLP_BS      |
+*     |     GLP_BS      |     GLP_NL      |      GLP_NU      |
+*     |     GLP_NL      |     GLP_BS      |      GLP_NL      |
+*     |     GLP_NL      |     GLP_NL      |      GLP_NL (*)  |
+*     +-----------------+-----------------+------------------+
+*
+*  Value of column q in solution to the original problem is the same as
+*  in solution to the transformed problem.
+*
+*  1. Formally, in solution to the transformed problem columns q and s
+*     cannot be non-basic at the same time, since the constraint (2)
+*     would be violated. However, if u[q] is close to zero, violation
+*     may be less than a working precision even if both columns q and s
+*     are non-basic. In this degenerate case row (2) can be only basic,
+*     i.e. non-active constraint (otherwise corresponding row of the
+*     basis matrix would be zero). This allows to pivot out auxiliary
+*     variable and pivot in column s, in which case the row becomes
+*     active while column s becomes basic.
+*
+*  2. If column q is integral, column s is also integral.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q in solution to the original problem is the same as
+*  in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q in solution to the original problem is the same as
+*  in solution to the transformed problem. */
+
+struct dbnd_col
+{     /* double-bounded column */
+      int q;
+      /* column reference number for variable x[q] */
+      int s;
+      /* column reference number for complement variable s */
+};
+
+static int rcv_dbnd_col(NPP *npp, void *info);
+
+void npp_dbnd_col(NPP *npp, NPPCOL *q)
+{     /* process non-negative column with upper bound */
+      struct dbnd_col *info;
+      NPPROW *p;
+      NPPCOL *s;
+      /* the column must be non-negative with upper bound */
+      xassert(q->lb == 0.0);
+      xassert(q->ub > 0.0);
+      xassert(q->ub != +DBL_MAX);
+      /* create variable s */
+      s = npp_add_col(npp);
+      s->is_int = q->is_int;
+      s->lb = 0.0, s->ub = +DBL_MAX;
+      /* create equality constraint (2) */
+      p = npp_add_row(npp);
+      p->lb = p->ub = q->ub;
+      npp_add_aij(npp, p, q, +1.0);
+      npp_add_aij(npp, p, s, +1.0);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_dbnd_col, sizeof(struct dbnd_col));
+      info->q = q->j;
+      info->s = s->j;
+      /* remove upper bound of x[q] */
+      q->ub = +DBL_MAX;
+      return;
+}
+
+static int rcv_dbnd_col(NPP *npp, void *_info)
+{     /* recover non-negative column with upper bound */
+      struct dbnd_col *info = _info;
+      if (npp->sol == GLP_BS)
+      {  if (npp->c_stat[info->q] == GLP_BS)
+         {  if (npp->c_stat[info->s] == GLP_BS)
+               npp->c_stat[info->q] = GLP_BS;
+            else if (npp->c_stat[info->s] == GLP_NL)
+               npp->c_stat[info->q] = GLP_NU;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else if (npp->c_stat[info->q] == GLP_NL)
+         {  if (npp->c_stat[info->s] == GLP_BS ||
+                npp->c_stat[info->s] == GLP_NL)
+               npp->c_stat[info->q] = GLP_NL;
+            else
+            {  npp_error();
+               return 1;
+            }
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_fixed_col - process fixed column
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_fixed_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_fixed_col processes column q, which is fixed:
+*
+*     x[q] = s[q],                                                   (1)
+*
+*  where s[q] is a fixed column value.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  The value of a fixed column can be substituted into the objective
+*  and constraint rows that allows removing the column from the problem.
+*
+*  Substituting x[q] = s[q] into the objective row, we have:
+*
+*     z = sum c[j] x[j] + c0 =
+*          j
+*
+*       = sum c[j] x[j] + c[q] x[q] + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] + c[q] s[q] + c0 =
+*         j!=q
+*
+*       = sum c[j] x[j] + c~0,
+*         j!=q
+*
+*  where
+*
+*     c~0 = c0 + c[q] s[q]                                           (2)
+*
+*  is the constant term of the objective in the transformed problem.
+*  Similarly, substituting x[q] = s[q] into constraint row i, we have:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i]  ==>
+*              j
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] x[q] <= U[i]  ==>
+*             j!=q
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] s[q] <= U[i]  ==>
+*             j!=q
+*
+*     L~[i] <= sum a[i,j] x[j] + a[i,q] s <= U~[i],
+*              j!=q
+*
+*  where
+*
+*     L~[i] = L[i] - a[i,q] s[q],  U~[i] = U[i] - a[i,q] s[q]        (3)
+*
+*  are lower and upper bounds of row i in the transformed problem,
+*  resp.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Column q is assigned status GLP_NS and its value is assigned s[q].
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q is assigned s[q].
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is assigned s[q]. */
+
+struct fixed_col
+{     /* fixed column */
+      int q;
+      /* column reference number for variable x[q] */
+      double s;
+      /* value, at which x[q] is fixed */
+};
+
+static int rcv_fixed_col(NPP *npp, void *info);
+
+void npp_fixed_col(NPP *npp, NPPCOL *q)
+{     /* process fixed column */
+      struct fixed_col *info;
+      NPPROW *i;
+      NPPAIJ *aij;
+      /* the column must be fixed */
+      xassert(q->lb == q->ub);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_fixed_col, sizeof(struct fixed_col));
+      info->q = q->j;
+      info->s = q->lb;
+      /* substitute x[q] = s[q] into objective row */
+      npp->c0 += q->coef * q->lb;
+      /* substitute x[q] = s[q] into constraint rows */
+      for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+      {  i = aij->row;
+         if (i->lb == i->ub)
+            i->ub = (i->lb -= aij->val * q->lb);
+         else
+         {  if (i->lb != -DBL_MAX)
+               i->lb -= aij->val * q->lb;
+            if (i->ub != +DBL_MAX)
+               i->ub -= aij->val * q->lb;
+         }
+      }
+      /* remove the column from the problem */
+      npp_del_col(npp, q);
+      return;
+}
+
+static int rcv_fixed_col(NPP *npp, void *_info)
+{     /* recover fixed column */
+      struct fixed_col *info = _info;
+      if (npp->sol == GLP_SOL)
+         npp->c_stat[info->q] = GLP_NS;
+      npp->c_value[info->q] = info->s;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_make_equality - process row with almost identical bounds
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_make_equality(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_make_equality processes row p:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],                               (1)
+*              j
+*
+*  where -oo < L[p] < U[p] < +oo, i.e. which is double-sided inequality
+*  constraint.
+*
+*  RETURNS
+*
+*  0 - row bounds have not been changed;
+*
+*  1 - row has been replaced by equality constraint.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  If bounds of row (1) are very close to each other:
+*
+*     U[p] - L[p] <= eps,                                            (2)
+*
+*  where eps is an absolute tolerance for row value, the row can be
+*  replaced by the following almost equivalent equiality constraint:
+*
+*     sum a[p,j] x[j] = b,                                           (3)
+*      j
+*
+*  where b = (L[p] + U[p]) / 2. If the right-hand side in (3) happens
+*  to be very close to its nearest integer:
+*
+*     |b - floor(b + 0.5)| <= eps,                                   (4)
+*
+*  it is reasonable to use this nearest integer as the right-hand side.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of row p in solution to the original problem is determined
+*  by its status and the sign of its multiplier pi[p] in solution to
+*  the transformed problem as follows:
+*
+*     +-----------------------+---------+--------------------+
+*     |    Status of row p    | Sign of |  Status of row p   |
+*     | (transformed problem) |  pi[p]  | (original problem) |
+*     +-----------------------+---------+--------------------+
+*     |        GLP_BS         |  + / -  |       GLP_BS       |
+*     |        GLP_NS         |    +    |       GLP_NL       |
+*     |        GLP_NS         |    -    |       GLP_NU       |
+*     +-----------------------+---------+--------------------+
+*
+*  Value of row multiplier pi[p] in solution to the original problem is
+*  the same as in solution to the transformed problem.
+*
+*  RECOVERING INTERIOR POINT SOLUTION
+*
+*  Value of row multiplier pi[p] in solution to the original problem is
+*  the same as in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct make_equality
+{     /* row with almost identical bounds */
+      int p;
+      /* row reference number */
+};
+
+static int rcv_make_equality(NPP *npp, void *info);
+
+int npp_make_equality(NPP *npp, NPPROW *p)
+{     /* process row with almost identical bounds */
+      struct make_equality *info;
+      double b, eps, nint;
+      /* the row must be double-sided inequality */
+      xassert(p->lb != -DBL_MAX);
+      xassert(p->ub != +DBL_MAX);
+      xassert(p->lb < p->ub);
+      /* check row bounds */
+      eps = 1e-9 + 1e-12 * fabs(p->lb);
+      if (p->ub - p->lb > eps) return 0;
+      /* row bounds are very close to each other */
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_make_equality, sizeof(struct make_equality));
+      info->p = p->i;
+      /* compute right-hand side */
+      b = 0.5 * (p->ub + p->lb);
+      nint = floor(b + 0.5);
+      if (fabs(b - nint) <= eps) b = nint;
+      /* replace row p by almost equivalent equality constraint */
+      p->lb = p->ub = b;
+      return 1;
+}
+
+int rcv_make_equality(NPP *npp, void *_info)
+{     /* recover row with almost identical bounds */
+      struct make_equality *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] == GLP_BS)
+            npp->r_stat[info->p] = GLP_BS;
+         else if (npp->r_stat[info->p] == GLP_NS)
+         {  if (npp->r_pi[info->p] >= 0.0)
+               npp->r_stat[info->p] = GLP_NL;
+            else
+               npp->r_stat[info->p] = GLP_NU;
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_make_fixed - process column with almost identical bounds
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_make_fixed(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_make_fixed processes column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where -oo < l[q] < u[q] < +oo, i.e. which has both lower and upper
+*  bounds.
+*
+*  RETURNS
+*
+*  0 - column bounds have not been changed;
+*
+*  1 - column has been fixed.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  If bounds of column (1) are very close to each other:
+*
+*     u[q] - l[q] <= eps,                                            (2)
+*
+*  where eps is an absolute tolerance for column value, the column can
+*  be fixed:
+*
+*     x[q] = s[q],                                                   (3)
+*
+*  where s[q] = (l[q] + u[q]) / 2. And if the fixed column value s[q]
+*  happens to be very close to its nearest integer:
+*
+*     |s[q] - floor(s[q] + 0.5)| <= eps,                             (4)
+*
+*  it is reasonable to use this nearest integer as the fixed value.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  In the dual system of the original (as well as transformed) problem
+*  column q corresponds to the following row:
+*
+*     sum a[i,q] pi[i] + lambda[q] = c[q].                           (5)
+*      i
+*
+*  Since multipliers pi[i] are known for all rows from solution to the
+*  transformed problem, formula (5) allows computing value of multiplier
+*  (reduced cost) for column q:
+*
+*     lambda[q] = c[q] - sum a[i,q] pi[i].                           (6)
+*                         i
+*
+*  Status of column q in solution to the original problem is determined
+*  by its status and the sign of its multiplier lambda[q] in solution to
+*  the transformed problem as follows:
+*
+*     +-----------------------+-----------+--------------------+
+*     |  Status of column q   |  Sign of  | Status of column q |
+*     | (transformed problem) | lambda[q] | (original problem) |
+*     +-----------------------+-----------+--------------------+
+*     |        GLP_BS         |   + / -   |       GLP_BS       |
+*     |        GLP_NS         |     +     |       GLP_NL       |
+*     |        GLP_NS         |     -     |       GLP_NU       |
+*     +-----------------------+-----------+--------------------+
+*
+*  Value of column q in solution to the original problem is the same as
+*  in solution to the transformed problem.
+*
+*  RECOVERING INTERIOR POINT SOLUTION
+*
+*  Value of column q in solution to the original problem is the same as
+*  in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct make_fixed
+{     /* column with almost identical bounds */
+      int q;
+      /* column reference number */
+      double c;
+      /* objective coefficient at x[q] */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[i,q] */
+};
+
+static int rcv_make_fixed(NPP *npp, void *info);
+
+int npp_make_fixed(NPP *npp, NPPCOL *q)
+{     /* process column with almost identical bounds */
+      struct make_fixed *info;
+      NPPAIJ *aij;
+      NPPLFE *lfe;
+      double s, eps, nint;
+      /* the column must be double-bounded */
+      xassert(q->lb != -DBL_MAX);
+      xassert(q->ub != +DBL_MAX);
+      xassert(q->lb < q->ub);
+      /* check column bounds */
+      eps = 1e-9 + 1e-12 * fabs(q->lb);
+      if (q->ub - q->lb > eps) return 0;
+      /* column bounds are very close to each other */
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_make_fixed, sizeof(struct make_fixed));
+      info->q = q->j;
+      info->c = q->coef;
+      info->ptr = NULL;
+      /* save column coefficients a[i,q] (needed for basic solution
+         only) */
+      if (npp->sol == GLP_SOL)
+      {  for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+         {  lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+            lfe->ref = aij->row->i;
+            lfe->val = aij->val;
+            lfe->next = info->ptr;
+            info->ptr = lfe;
+         }
+      }
+      /* compute column fixed value */
+      s = 0.5 * (q->ub + q->lb);
+      nint = floor(s + 0.5);
+      if (fabs(s - nint) <= eps) s = nint;
+      /* make column q fixed */
+      q->lb = q->ub = s;
+      return 1;
+}
+
+static int rcv_make_fixed(NPP *npp, void *_info)
+{     /* recover column with almost identical bounds */
+      struct make_fixed *info = _info;
+      NPPLFE *lfe;
+      double lambda;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->c_stat[info->q] == GLP_BS)
+            npp->c_stat[info->q] = GLP_BS;
+         else if (npp->c_stat[info->q] == GLP_NS)
+         {  /* compute multiplier for column q with formula (6) */
+            lambda = info->c;
+            for (lfe = info->ptr; lfe != NULL; lfe = lfe->next)
+               lambda -= lfe->val * npp->r_pi[lfe->ref];
+            /* assign status to non-basic column */
+            if (lambda >= 0.0)
+               npp->c_stat[info->q] = GLP_NL;
+            else
+               npp->c_stat[info->q] = GLP_NU;
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp3.c b/test/monniaux/glpk-4.65/src/npp/npp3.c
new file mode 100644
index 00000000..883af127
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp3.c
@@ -0,0 +1,2861 @@
+/* npp3.c */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+/***********************************************************************
+*  NAME
+*
+*  npp_empty_row - process empty row
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_empty_row(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_empty_row processes row p, which is empty, i.e.
+*  coefficients at all columns in this row are zero:
+*
+*     L[p] <= sum 0 x[j] <= U[p],                                    (1)
+*
+*  where L[p] <= U[p].
+*
+*  RETURNS
+*
+*  0 - success;
+*
+*  1 - problem has no primal feasible solution.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  If the following conditions hold:
+*
+*     L[p] <= +eps,  U[p] >= -eps,                                   (2)
+*
+*  where eps is an absolute tolerance for row value, the row p is
+*  redundant. In this case it can be replaced by equivalent redundant
+*  row, which is free (unbounded), and then removed from the problem.
+*  Otherwise, the row p is infeasible and, thus, the problem has no
+*  primal feasible solution.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  See the routine npp_free_row.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  See the routine npp_free_row.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+int npp_empty_row(NPP *npp, NPPROW *p)
+{     /* process empty row */
+      double eps = 1e-3;
+      /* the row must be empty */
+      xassert(p->ptr == NULL);
+      /* check primal feasibility */
+      if (p->lb > +eps || p->ub < -eps)
+         return 1;
+      /* replace the row by equivalent free (unbounded) row */
+      p->lb = -DBL_MAX, p->ub = +DBL_MAX;
+      /* and process it */
+      npp_free_row(npp, p);
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_empty_col - process empty column
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_empty_col(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_empty_col processes column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] <= u[q], which is empty, i.e. has zero coefficients in
+*  all constraint rows.
+*
+*  RETURNS
+*
+*  0 - success;
+*
+*  1 - problem has no dual feasible solution.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  The row of the dual system corresponding to the empty column is the
+*  following:
+*
+*     sum 0 pi[i] + lambda[q] = c[q],                                (2)
+*      i
+*
+*  from which it follows that:
+*
+*     lambda[q] = c[q].                                              (3)
+*
+*  If the following condition holds:
+*
+*     c[q] < - eps,                                                  (4)
+*
+*  where eps is an absolute tolerance for column multiplier, the lower
+*  column bound l[q] must be active to provide dual feasibility (note
+*  that being preprocessed the problem is always minimization). In this
+*  case the column can be fixed on its lower bound and removed from the
+*  problem (if the column is integral, its bounds are also assumed to
+*  be integral). And if the column has no lower bound (l[q] = -oo), the
+*  problem has no dual feasible solution.
+*
+*  If the following condition holds:
+*
+*     c[q] > + eps,                                                  (5)
+*
+*  the upper column bound u[q] must be active to provide dual
+*  feasibility. In this case the column can be fixed on its upper bound
+*  and removed from the problem. And if the column has no upper bound
+*  (u[q] = +oo), the problem has no dual feasible solution.
+*
+*  Finally, if the following condition holds:
+*
+*     - eps <= c[q] <= +eps,                                         (6)
+*
+*  dual feasibility does not depend on a particular value of column q.
+*  In this case the column can be fixed either on its lower bound (if
+*  l[q] > -oo) or on its upper bound (if u[q] < +oo) or at zero (if the
+*  column is unbounded) and then removed from the problem.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  See the routine npp_fixed_col. Having been recovered the column
+*  is assigned status GLP_NS. However, if actually it is not fixed
+*  (l[q] < u[q]), its status should be changed to GLP_NL, GLP_NU, or
+*  GLP_NF depending on which bound it was fixed on transformation stage.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  See the routine npp_fixed_col.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  See the routine npp_fixed_col. */
+
+struct empty_col
+{     /* empty column */
+      int q;
+      /* column reference number */
+      char stat;
+      /* status in basic solution */
+};
+
+static int rcv_empty_col(NPP *npp, void *info);
+
+int npp_empty_col(NPP *npp, NPPCOL *q)
+{     /* process empty column */
+      struct empty_col *info;
+      double eps = 1e-3;
+      /* the column must be empty */
+      xassert(q->ptr == NULL);
+      /* check dual feasibility */
+      if (q->coef > +eps && q->lb == -DBL_MAX)
+         return 1;
+      if (q->coef < -eps && q->ub == +DBL_MAX)
+         return 1;
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_empty_col, sizeof(struct empty_col));
+      info->q = q->j;
+      /* fix the column */
+      if (q->lb == -DBL_MAX && q->ub == +DBL_MAX)
+      {  /* free column */
+         info->stat = GLP_NF;
+         q->lb = q->ub = 0.0;
+      }
+      else if (q->ub == +DBL_MAX)
+lo:   {  /* column with lower bound */
+         info->stat = GLP_NL;
+         q->ub = q->lb;
+      }
+      else if (q->lb == -DBL_MAX)
+up:   {  /* column with upper bound */
+         info->stat = GLP_NU;
+         q->lb = q->ub;
+      }
+      else if (q->lb != q->ub)
+      {  /* double-bounded column */
+         if (q->coef >= +DBL_EPSILON) goto lo;
+         if (q->coef <= -DBL_EPSILON) goto up;
+         if (fabs(q->lb) <= fabs(q->ub)) goto lo; else goto up;
+      }
+      else
+      {  /* fixed column */
+         info->stat = GLP_NS;
+      }
+      /* process fixed column */
+      npp_fixed_col(npp, q);
+      return 0;
+}
+
+static int rcv_empty_col(NPP *npp, void *_info)
+{     /* recover empty column */
+      struct empty_col *info = _info;
+      if (npp->sol == GLP_SOL)
+         npp->c_stat[info->q] = info->stat;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_value - process implied column value
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_implied_value(NPP *npp, NPPCOL *q, double s);
+*
+*  DESCRIPTION
+*
+*  For column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] < u[q], the routine npp_implied_value processes its
+*  implied value s[q]. If this implied value satisfies to the current
+*  column bounds and integrality condition, the routine fixes column q
+*  at the given point. Note that the column is kept in the problem in
+*  any case.
+*
+*  RETURNS
+*
+*  0 - column has been fixed;
+*
+*  1 - implied value violates to current column bounds;
+*
+*  2 - implied value violates integrality condition.
+*
+*  ALGORITHM
+*
+*  Implied column value s[q] satisfies to the current column bounds if
+*  the following condition holds:
+*
+*     l[q] - eps <= s[q] <= u[q] + eps,                              (2)
+*
+*  where eps is an absolute tolerance for column value. If the column
+*  is integral, the following condition also must hold:
+*
+*     |s[q] - floor(s[q]+0.5)| <= eps,                               (3)
+*
+*  where floor(s[q]+0.5) is the nearest integer to s[q].
+*
+*  If both condition (2) and (3) are satisfied, the column can be fixed
+*  at the value s[q], or, if it is integral, at floor(s[q]+0.5).
+*  Otherwise, if s[q] violates (2) or (3), the problem has no feasible
+*  solution.
+*
+*  Note: If s[q] is close to l[q] or u[q], it seems to be reasonable to
+*  fix the column at its lower or upper bound, resp. rather than at the
+*  implied value. */
+
+int npp_implied_value(NPP *npp, NPPCOL *q, double s)
+{     /* process implied column value */
+      double eps, nint;
+      xassert(npp == npp);
+      /* column must not be fixed */
+      xassert(q->lb < q->ub);
+      /* check integrality */
+      if (q->is_int)
+      {  nint = floor(s + 0.5);
+         if (fabs(s - nint) <= 1e-5)
+            s = nint;
+         else
+            return 2;
+      }
+      /* check current column lower bound */
+      if (q->lb != -DBL_MAX)
+      {  eps = (q->is_int ? 1e-5 : 1e-5 + 1e-8 * fabs(q->lb));
+         if (s < q->lb - eps) return 1;
+         /* if s[q] is close to l[q], fix column at its lower bound
+            rather than at the implied value */
+         if (s < q->lb + 1e-3 * eps)
+         {  q->ub = q->lb;
+            return 0;
+         }
+      }
+      /* check current column upper bound */
+      if (q->ub != +DBL_MAX)
+      {  eps = (q->is_int ? 1e-5 : 1e-5 + 1e-8 * fabs(q->ub));
+         if (s > q->ub + eps) return 1;
+         /* if s[q] is close to u[q], fix column at its upper bound
+            rather than at the implied value */
+         if (s > q->ub - 1e-3 * eps)
+         {  q->lb = q->ub;
+            return 0;
+         }
+      }
+      /* fix column at the implied value */
+      q->lb = q->ub = s;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_eq_singlet - process row singleton (equality constraint)
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_eq_singlet(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_eq_singlet processes row p, which is equiality
+*  constraint having the only non-zero coefficient:
+*
+*     a[p,q] x[q] = b.                                               (1)
+*
+*  RETURNS
+*
+*  0 - success;
+*
+*  1 - problem has no primal feasible solution;
+*
+*  2 - problem has no integer feasible solution.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  The equality constraint defines implied value of column q:
+*
+*     x[q] = s[q] = b / a[p,q].                                      (2)
+*
+*  If the implied value s[q] satisfies to the column bounds (see the
+*  routine npp_implied_value), the column can be fixed at s[q] and
+*  removed from the problem. In this case row p becomes redundant, so
+*  it can be replaced by equivalent free row and also removed from the
+*  problem.
+*
+*  Note that the routine removes from the problem only row p. Column q
+*  becomes fixed, however, it is kept in the problem.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  In solution to the original problem row p is assigned status GLP_NS
+*  (active equality constraint), and column q is assigned status GLP_BS
+*  (basic column).
+*
+*  Multiplier for row p can be computed as follows. In the dual system
+*  of the original problem column q corresponds to the following row:
+*
+*     sum a[i,q] pi[i] + lambda[q] = c[q]  ==>
+*      i
+*
+*     sum a[i,q] pi[i] + a[p,q] pi[p] + lambda[q] = c[q].
+*     i!=p
+*
+*  Therefore:
+*
+*               1
+*     pi[p] = ------ (c[q] - lambda[q] - sum a[i,q] pi[i]),          (3)
+*             a[p,q]                     i!=q
+*
+*  where lambda[q] = 0 (since column[q] is basic), and pi[i] for all
+*  i != p are known in solution to the transformed problem.
+*
+*  Value of column q in solution to the original problem is assigned
+*  its implied value s[q].
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Multiplier for row p is computed with formula (3). Value of column
+*  q is assigned its implied value s[q].
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is assigned its implied value s[q]. */
+
+struct eq_singlet
+{     /* row singleton (equality constraint) */
+      int p;
+      /* row reference number */
+      int q;
+      /* column reference number */
+      double apq;
+      /* constraint coefficient a[p,q] */
+      double c;
+      /* objective coefficient at x[q] */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[i,q], i != p */
+};
+
+static int rcv_eq_singlet(NPP *npp, void *info);
+
+int npp_eq_singlet(NPP *npp, NPPROW *p)
+{     /* process row singleton (equality constraint) */
+      struct eq_singlet *info;
+      NPPCOL *q;
+      NPPAIJ *aij;
+      NPPLFE *lfe;
+      int ret;
+      double s;
+      /* the row must be singleton equality constraint */
+      xassert(p->lb == p->ub);
+      xassert(p->ptr != NULL && p->ptr->r_next == NULL);
+      /* compute and process implied column value */
+      aij = p->ptr;
+      q = aij->col;
+      s = p->lb / aij->val;
+      ret = npp_implied_value(npp, q, s);
+      xassert(0 <= ret && ret <= 2);
+      if (ret != 0) return ret;
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_eq_singlet, sizeof(struct eq_singlet));
+      info->p = p->i;
+      info->q = q->j;
+      info->apq = aij->val;
+      info->c = q->coef;
+      info->ptr = NULL;
+      /* save column coefficients a[i,q], i != p (not needed for MIP
+         solution) */
+      if (npp->sol != GLP_MIP)
+      {  for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+         {  if (aij->row == p) continue; /* skip a[p,q] */
+            lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+            lfe->ref = aij->row->i;
+            lfe->val = aij->val;
+            lfe->next = info->ptr;
+            info->ptr = lfe;
+         }
+      }
+      /* remove the row from the problem */
+      npp_del_row(npp, p);
+      return 0;
+}
+
+static int rcv_eq_singlet(NPP *npp, void *_info)
+{     /* recover row singleton (equality constraint) */
+      struct eq_singlet *info = _info;
+      NPPLFE *lfe;
+      double temp;
+      if (npp->sol == GLP_SOL)
+      {  /* column q must be already recovered as GLP_NS */
+         if (npp->c_stat[info->q] != GLP_NS)
+         {  npp_error();
+            return 1;
+         }
+         npp->r_stat[info->p] = GLP_NS;
+         npp->c_stat[info->q] = GLP_BS;
+      }
+      if (npp->sol != GLP_MIP)
+      {  /* compute multiplier for row p with formula (3) */
+         temp = info->c;
+         for (lfe = info->ptr; lfe != NULL; lfe = lfe->next)
+            temp -= lfe->val * npp->r_pi[lfe->ref];
+         npp->r_pi[info->p] = temp / info->apq;
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_lower - process implied column lower bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_implied_lower(NPP *npp, NPPCOL *q, double l);
+*
+*  DESCRIPTION
+*
+*  For column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] < u[q], the routine npp_implied_lower processes its
+*  implied lower bound l'[q]. As the result the current column lower
+*  bound may increase. Note that the column is kept in the problem in
+*  any case.
+*
+*  RETURNS
+*
+*  0 - current column lower bound has not changed;
+*
+*  1 - current column lower bound has changed, but not significantly;
+*
+*  2 - current column lower bound has significantly changed;
+*
+*  3 - column has been fixed on its upper bound;
+*
+*  4 - implied lower bound violates current column upper bound.
+*
+*  ALGORITHM
+*
+*  If column q is integral, before processing its implied lower bound
+*  should be rounded up:
+*
+*              ( floor(l'[q]+0.5), if |l'[q] - floor(l'[q]+0.5)| <= eps
+*     l'[q] := <                                                     (2)
+*              ( ceil(l'[q]),      otherwise
+*
+*  where floor(l'[q]+0.5) is the nearest integer to l'[q], ceil(l'[q])
+*  is smallest integer not less than l'[q], and eps is an absolute
+*  tolerance for column value.
+*
+*  Processing implied column lower bound l'[q] includes the following
+*  cases:
+*
+*  1) if l'[q] < l[q] + eps, implied lower bound is redundant;
+*
+*  2) if l[q] + eps <= l[q] <= u[q] + eps, current column lower bound
+*     l[q] can be strengthened by replacing it with l'[q]. If in this
+*     case new column lower bound becomes close to current column upper
+*     bound u[q], the column can be fixed on its upper bound;
+*
+*  3) if l'[q] > u[q] + eps, implied lower bound violates current
+*     column upper bound u[q], in which case the problem has no primal
+*     feasible solution. */
+
+int npp_implied_lower(NPP *npp, NPPCOL *q, double l)
+{     /* process implied column lower bound */
+      int ret;
+      double eps, nint;
+      xassert(npp == npp);
+      /* column must not be fixed */
+      xassert(q->lb < q->ub);
+      /* implied lower bound must be finite */
+      xassert(l != -DBL_MAX);
+      /* if column is integral, round up l'[q] */
+      if (q->is_int)
+      {  nint = floor(l + 0.5);
+         if (fabs(l - nint) <= 1e-5)
+            l = nint;
+         else
+            l = ceil(l);
+      }
+      /* check current column lower bound */
+      if (q->lb != -DBL_MAX)
+      {  eps = (q->is_int ? 1e-3 : 1e-3 + 1e-6 * fabs(q->lb));
+         if (l < q->lb + eps)
+         {  ret = 0; /* redundant */
+            goto done;
+         }
+      }
+      /* check current column upper bound */
+      if (q->ub != +DBL_MAX)
+      {  eps = (q->is_int ? 1e-5 : 1e-5 + 1e-8 * fabs(q->ub));
+         if (l > q->ub + eps)
+         {  ret = 4; /* infeasible */
+            goto done;
+         }
+         /* if l'[q] is close to u[q], fix column at its upper bound */
+         if (l > q->ub - 1e-3 * eps)
+         {  q->lb = q->ub;
+            ret = 3; /* fixed */
+            goto done;
+         }
+      }
+      /* check if column lower bound changes significantly */
+      if (q->lb == -DBL_MAX)
+         ret = 2; /* significantly */
+      else if (q->is_int && l > q->lb + 0.5)
+         ret = 2; /* significantly */
+      else if (l > q->lb + 0.30 * (1.0 + fabs(q->lb)))
+         ret = 2; /* significantly */
+      else
+         ret = 1; /* not significantly */
+      /* set new column lower bound */
+      q->lb = l;
+done: return ret;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_upper - process implied column upper bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_implied_upper(NPP *npp, NPPCOL *q, double u);
+*
+*  DESCRIPTION
+*
+*  For column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] < u[q], the routine npp_implied_upper processes its
+*  implied upper bound u'[q]. As the result the current column upper
+*  bound may decrease. Note that the column is kept in the problem in
+*  any case.
+*
+*  RETURNS
+*
+*  0 - current column upper bound has not changed;
+*
+*  1 - current column upper bound has changed, but not significantly;
+*
+*  2 - current column upper bound has significantly changed;
+*
+*  3 - column has been fixed on its lower bound;
+*
+*  4 - implied upper bound violates current column lower bound.
+*
+*  ALGORITHM
+*
+*  If column q is integral, before processing its implied upper bound
+*  should be rounded down:
+*
+*              ( floor(u'[q]+0.5), if |u'[q] - floor(l'[q]+0.5)| <= eps
+*     u'[q] := <                                                     (2)
+*              ( floor(l'[q]),     otherwise
+*
+*  where floor(u'[q]+0.5) is the nearest integer to u'[q],
+*  floor(u'[q]) is largest integer not greater than u'[q], and eps is
+*  an absolute tolerance for column value.
+*
+*  Processing implied column upper bound u'[q] includes the following
+*  cases:
+*
+*  1) if u'[q] > u[q] - eps, implied upper bound is redundant;
+*
+*  2) if l[q] - eps <= u[q] <= u[q] - eps, current column upper bound
+*     u[q] can be strengthened by replacing it with u'[q]. If in this
+*     case new column upper bound becomes close to current column lower
+*     bound, the column can be fixed on its lower bound;
+*
+*  3) if u'[q] < l[q] - eps, implied upper bound violates current
+*     column lower bound l[q], in which case the problem has no primal
+*     feasible solution. */
+
+int npp_implied_upper(NPP *npp, NPPCOL *q, double u)
+{     int ret;
+      double eps, nint;
+      xassert(npp == npp);
+      /* column must not be fixed */
+      xassert(q->lb < q->ub);
+      /* implied upper bound must be finite */
+      xassert(u != +DBL_MAX);
+      /* if column is integral, round down u'[q] */
+      if (q->is_int)
+      {  nint = floor(u + 0.5);
+         if (fabs(u - nint) <= 1e-5)
+            u = nint;
+         else
+            u = floor(u);
+      }
+      /* check current column upper bound */
+      if (q->ub != +DBL_MAX)
+      {  eps = (q->is_int ? 1e-3 : 1e-3 + 1e-6 * fabs(q->ub));
+         if (u > q->ub - eps)
+         {  ret = 0; /* redundant */
+            goto done;
+         }
+      }
+      /* check current column lower bound */
+      if (q->lb != -DBL_MAX)
+      {  eps = (q->is_int ? 1e-5 : 1e-5 + 1e-8 * fabs(q->lb));
+         if (u < q->lb - eps)
+         {  ret = 4; /* infeasible */
+            goto done;
+         }
+         /* if u'[q] is close to l[q], fix column at its lower bound */
+         if (u < q->lb + 1e-3 * eps)
+         {  q->ub = q->lb;
+            ret = 3; /* fixed */
+            goto done;
+         }
+      }
+      /* check if column upper bound changes significantly */
+      if (q->ub == +DBL_MAX)
+         ret = 2; /* significantly */
+      else if (q->is_int && u < q->ub - 0.5)
+         ret = 2; /* significantly */
+      else if (u < q->ub - 0.30 * (1.0 + fabs(q->ub)))
+         ret = 2; /* significantly */
+      else
+         ret = 1; /* not significantly */
+      /* set new column upper bound */
+      q->ub = u;
+done: return ret;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_ineq_singlet - process row singleton (inequality constraint)
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_ineq_singlet(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_ineq_singlet processes row p, which is inequality
+*  constraint having the only non-zero coefficient:
+*
+*     L[p] <= a[p,q] * x[q] <= U[p],                                 (1)
+*
+*  where L[p] < U[p], L[p] > -oo and/or U[p] < +oo.
+*
+*  RETURNS
+*
+*  0 - current column bounds have not changed;
+*
+*  1 - current column bounds have changed, but not significantly;
+*
+*  2 - current column bounds have significantly changed;
+*
+*  3 - column has been fixed on its lower or upper bound;
+*
+*  4 - problem has no primal feasible solution.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Inequality constraint (1) defines implied bounds of column q:
+*
+*             (  L[p] / a[p,q],  if a[p,q] > 0
+*     l'[q] = <                                                      (2)
+*             (  U[p] / a[p,q],  if a[p,q] < 0
+*
+*             (  U[p] / a[p,q],  if a[p,q] > 0
+*     u'[q] = <                                                      (3)
+*             (  L[p] / a[p,q],  if a[p,q] < 0
+*
+*  If these implied bounds do not violate current bounds of column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (4)
+*
+*  they can be used to strengthen the current column bounds:
+*
+*     l[q] := max(l[q], l'[q]),                                      (5)
+*
+*     u[q] := min(u[q], u'[q]).                                      (6)
+*
+*  (See the routines npp_implied_lower and npp_implied_upper.)
+*
+*  Once bounds of row p (1) have been carried over column q, the row
+*  becomes redundant, so it can be replaced by equivalent free row and
+*  removed from the problem.
+*
+*  Note that the routine removes from the problem only row p. Column q,
+*  even it has been fixed, is kept in the problem.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Note that the row in the dual system corresponding to column q is
+*  the following:
+*
+*     sum a[i,q] pi[i] + lambda[q] = c[q]  ==>
+*      i
+*                                                                    (7)
+*     sum a[i,q] pi[i] + a[p,q] pi[p] + lambda[q] = c[q],
+*     i!=p
+*
+*  where pi[i] for all i != p are known in solution to the transformed
+*  problem. Row p does not exist in the transformed problem, so it has
+*  zero multiplier there. This allows computing multiplier for column q
+*  in solution to the transformed problem:
+*
+*     lambda~[q] = c[q] - sum a[i,q] pi[i].                          (8)
+*                         i!=p
+*
+*  Let in solution to the transformed problem column q be non-basic
+*  with lower bound active (GLP_NL, lambda~[q] >= 0), and this lower
+*  bound be implied one l'[q]. From the original problem's standpoint
+*  this then means that actually the original column lower bound l[q]
+*  is inactive, and active is that row bound L[p] or U[p] that defines
+*  the implied bound l'[q] (2). In this case in solution to the
+*  original problem column q is assigned status GLP_BS while row p is
+*  assigned status GLP_NL (if a[p,q] > 0) or GLP_NU (if a[p,q] < 0).
+*  Since now column q is basic, its multiplier lambda[q] is zero. This
+*  allows using (7) and (8) to find multiplier for row p in solution to
+*  the original problem:
+*
+*               1
+*     pi[p] = ------ (c[q] - sum a[i,q] pi[i]) = lambda~[q] / a[p,q] (9)
+*             a[p,q]         i!=p
+*
+*  Now let in solution to the transformed problem column q be non-basic
+*  with upper bound active (GLP_NU, lambda~[q] <= 0), and this upper
+*  bound be implied one u'[q]. As in the previous case this then means
+*  that from the original problem's standpoint actually the original
+*  column upper bound u[q] is inactive, and active is that row bound
+*  L[p] or U[p] that defines the implied bound u'[q] (3). In this case
+*  in solution to the original problem column q is assigned status
+*  GLP_BS, row p is assigned status GLP_NU (if a[p,q] > 0) or GLP_NL
+*  (if a[p,q] < 0), and its multiplier is computed with formula (9).
+*
+*  Strengthening bounds of column q according to (5) and (6) may make
+*  it fixed. Thus, if in solution to the transformed problem column q is
+*  non-basic and fixed (GLP_NS), we can suppose that if lambda~[q] > 0,
+*  column q has active lower bound (GLP_NL), and if lambda~[q] < 0,
+*  column q has active upper bound (GLP_NU), reducing this case to two
+*  previous ones. If, however, lambda~[q] is close to zero or
+*  corresponding bound of row p does not exist (this may happen if
+*  lambda~[q] has wrong sign due to round-off errors, in which case it
+*  is expected to be close to zero, since solution is assumed to be dual
+*  feasible), column q can be assigned status GLP_BS (basic), and row p
+*  can be made active on its existing bound. In the latter case row
+*  multiplier pi[p] computed with formula (9) will be also close to
+*  zero, and dual feasibility will be kept.
+*
+*  In all other cases, namely, if in solution to the transformed
+*  problem column q is basic (GLP_BS), or non-basic with original lower
+*  bound l[q] active (GLP_NL), or non-basic with original upper bound
+*  u[q] active (GLP_NU), constraint (1) is inactive. So in solution to
+*  the original problem status of column q remains unchanged, row p is
+*  assigned status GLP_BS, and its multiplier pi[p] is assigned zero
+*  value.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  First, value of multiplier for column q in solution to the original
+*  problem is computed with formula (8). If lambda~[q] > 0 and column q
+*  has implied lower bound, or if lambda~[q] < 0 and column q has
+*  implied upper bound, this means that from the original problem's
+*  standpoint actually row p has corresponding active bound, in which
+*  case its multiplier pi[p] is computed with formula (9). In other
+*  cases, when the sign of lambda~[q] corresponds to original bound of
+*  column q, or when lambda~[q] =~ 0, value of row multiplier pi[p] is
+*  assigned zero value.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct ineq_singlet
+{     /* row singleton (inequality constraint) */
+      int p;
+      /* row reference number */
+      int q;
+      /* column reference number */
+      double apq;
+      /* constraint coefficient a[p,q] */
+      double c;
+      /* objective coefficient at x[q] */
+      double lb;
+      /* row lower bound */
+      double ub;
+      /* row upper bound */
+      char lb_changed;
+      /* this flag is set if column lower bound was changed */
+      char ub_changed;
+      /* this flag is set if column upper bound was changed */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[i,q], i != p */
+};
+
+static int rcv_ineq_singlet(NPP *npp, void *info);
+
+int npp_ineq_singlet(NPP *npp, NPPROW *p)
+{     /* process row singleton (inequality constraint) */
+      struct ineq_singlet *info;
+      NPPCOL *q;
+      NPPAIJ *apq, *aij;
+      NPPLFE *lfe;
+      int lb_changed, ub_changed;
+      double ll, uu;
+      /* the row must be singleton inequality constraint */
+      xassert(p->lb != -DBL_MAX || p->ub != +DBL_MAX);
+      xassert(p->lb < p->ub);
+      xassert(p->ptr != NULL && p->ptr->r_next == NULL);
+      /* compute implied column bounds */
+      apq = p->ptr;
+      q = apq->col;
+      xassert(q->lb < q->ub);
+      if (apq->val > 0.0)
+      {  ll = (p->lb == -DBL_MAX ? -DBL_MAX : p->lb / apq->val);
+         uu = (p->ub == +DBL_MAX ? +DBL_MAX : p->ub / apq->val);
+      }
+      else
+      {  ll = (p->ub == +DBL_MAX ? -DBL_MAX : p->ub / apq->val);
+         uu = (p->lb == -DBL_MAX ? +DBL_MAX : p->lb / apq->val);
+      }
+      /* process implied column lower bound */
+      if (ll == -DBL_MAX)
+         lb_changed = 0;
+      else
+      {  lb_changed = npp_implied_lower(npp, q, ll);
+         xassert(0 <= lb_changed && lb_changed <= 4);
+         if (lb_changed == 4) return 4; /* infeasible */
+      }
+      /* process implied column upper bound */
+      if (uu == +DBL_MAX)
+         ub_changed = 0;
+      else if (lb_changed == 3)
+      {  /* column was fixed on its upper bound due to l'[q] = u[q] */
+         /* note that L[p] < U[p], so l'[q] = u[q] < u'[q] */
+         ub_changed = 0;
+      }
+      else
+      {  ub_changed = npp_implied_upper(npp, q, uu);
+         xassert(0 <= ub_changed && ub_changed <= 4);
+         if (ub_changed == 4) return 4; /* infeasible */
+      }
+      /* if neither lower nor upper column bound was changed, the row
+         is originally redundant and can be replaced by free row */
+      if (!lb_changed && !ub_changed)
+      {  p->lb = -DBL_MAX, p->ub = +DBL_MAX;
+         npp_free_row(npp, p);
+         return 0;
+      }
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_ineq_singlet, sizeof(struct ineq_singlet));
+      info->p = p->i;
+      info->q = q->j;
+      info->apq = apq->val;
+      info->c = q->coef;
+      info->lb = p->lb;
+      info->ub = p->ub;
+      info->lb_changed = (char)lb_changed;
+      info->ub_changed = (char)ub_changed;
+      info->ptr = NULL;
+      /* save column coefficients a[i,q], i != p (not needed for MIP
+         solution) */
+      if (npp->sol != GLP_MIP)
+      {  for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+         {  if (aij == apq) continue; /* skip a[p,q] */
+            lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+            lfe->ref = aij->row->i;
+            lfe->val = aij->val;
+            lfe->next = info->ptr;
+            info->ptr = lfe;
+         }
+      }
+      /* remove the row from the problem */
+      npp_del_row(npp, p);
+      return lb_changed >= ub_changed ? lb_changed : ub_changed;
+}
+
+static int rcv_ineq_singlet(NPP *npp, void *_info)
+{     /* recover row singleton (inequality constraint) */
+      struct ineq_singlet *info = _info;
+      NPPLFE *lfe;
+      double lambda;
+      if (npp->sol == GLP_MIP) goto done;
+      /* compute lambda~[q] in solution to the transformed problem
+         with formula (8) */
+      lambda = info->c;
+      for (lfe = info->ptr; lfe != NULL; lfe = lfe->next)
+         lambda -= lfe->val * npp->r_pi[lfe->ref];
+      if (npp->sol == GLP_SOL)
+      {  /* recover basic solution */
+         if (npp->c_stat[info->q] == GLP_BS)
+         {  /* column q is basic, so row p is inactive */
+            npp->r_stat[info->p] = GLP_BS;
+            npp->r_pi[info->p] = 0.0;
+         }
+         else if (npp->c_stat[info->q] == GLP_NL)
+nl:      {  /* column q is non-basic with lower bound active */
+            if (info->lb_changed)
+            {  /* it is implied bound, so actually row p is active
+                  while column q is basic */
+               npp->r_stat[info->p] =
+                  (char)(info->apq > 0.0 ? GLP_NL : GLP_NU);
+               npp->c_stat[info->q] = GLP_BS;
+               npp->r_pi[info->p] = lambda / info->apq;
+            }
+            else
+            {  /* it is original bound, so row p is inactive */
+               npp->r_stat[info->p] = GLP_BS;
+               npp->r_pi[info->p] = 0.0;
+            }
+         }
+         else if (npp->c_stat[info->q] == GLP_NU)
+nu:      {  /* column q is non-basic with upper bound active */
+            if (info->ub_changed)
+            {  /* it is implied bound, so actually row p is active
+                  while column q is basic */
+               npp->r_stat[info->p] =
+                  (char)(info->apq > 0.0 ? GLP_NU : GLP_NL);
+               npp->c_stat[info->q] = GLP_BS;
+               npp->r_pi[info->p] = lambda / info->apq;
+            }
+            else
+            {  /* it is original bound, so row p is inactive */
+               npp->r_stat[info->p] = GLP_BS;
+               npp->r_pi[info->p] = 0.0;
+            }
+         }
+         else if (npp->c_stat[info->q] == GLP_NS)
+         {  /* column q is non-basic and fixed; note, however, that in
+               in the original problem it is non-fixed */
+            if (lambda > +1e-7)
+            {  if (info->apq > 0.0 && info->lb != -DBL_MAX ||
+                   info->apq < 0.0 && info->ub != +DBL_MAX ||
+                  !info->lb_changed)
+               {  /* either corresponding bound of row p exists or
+                     column q remains non-basic with its original lower
+                     bound active */
+                  npp->c_stat[info->q] = GLP_NL;
+                  goto nl;
+               }
+            }
+            if (lambda < -1e-7)
+            {  if (info->apq > 0.0 && info->ub != +DBL_MAX ||
+                   info->apq < 0.0 && info->lb != -DBL_MAX ||
+                  !info->ub_changed)
+               {  /* either corresponding bound of row p exists or
+                     column q remains non-basic with its original upper
+                     bound active */
+                  npp->c_stat[info->q] = GLP_NU;
+                  goto nu;
+               }
+            }
+            /* either lambda~[q] is close to zero, or corresponding
+               bound of row p does not exist, because lambda~[q] has
+               wrong sign due to round-off errors; in the latter case
+               lambda~[q] is also assumed to be close to zero; so, we
+               can make row p active on its existing bound and column q
+               basic; pi[p] will have wrong sign, but it also will be
+               close to zero (rarus casus of dual degeneracy) */
+            if (info->lb != -DBL_MAX && info->ub == +DBL_MAX)
+            {  /* row lower bound exists, but upper bound doesn't */
+               npp->r_stat[info->p] = GLP_NL;
+            }
+            else if (info->lb == -DBL_MAX && info->ub != +DBL_MAX)
+            {  /* row upper bound exists, but lower bound doesn't */
+               npp->r_stat[info->p] = GLP_NU;
+            }
+            else if (info->lb != -DBL_MAX && info->ub != +DBL_MAX)
+            {  /* both row lower and upper bounds exist */
+               /* to choose proper active row bound we should not use
+                  lambda~[q], because its value being close to zero is
+                  unreliable; so we choose that bound which provides
+                  primal feasibility for original constraint (1) */
+               if (info->apq * npp->c_value[info->q] <=
+                   0.5 * (info->lb + info->ub))
+                  npp->r_stat[info->p] = GLP_NL;
+               else
+                  npp->r_stat[info->p] = GLP_NU;
+            }
+            else
+            {  npp_error();
+               return 1;
+            }
+            npp->c_stat[info->q] = GLP_BS;
+            npp->r_pi[info->p] = lambda / info->apq;
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      if (npp->sol == GLP_IPT)
+      {  /* recover interior-point solution */
+         if (lambda > +DBL_EPSILON && info->lb_changed ||
+             lambda < -DBL_EPSILON && info->ub_changed)
+         {  /* actually row p has corresponding active bound */
+            npp->r_pi[info->p] = lambda / info->apq;
+         }
+         else
+         {  /* either bounds of column q are both inactive or its
+               original bound is active */
+            npp->r_pi[info->p] = 0.0;
+         }
+      }
+done: return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_slack - process column singleton (implied slack variable)
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_implied_slack(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_implied_slack processes column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] < u[q], having the only non-zero coefficient in row p,
+*  which is equality constraint:
+*
+*     sum a[p,j] x[j] + a[p,q] x[q] = b.                             (2)
+*     j!=q
+*
+*  PROBLEM TRANSFORMATION
+*
+*  (If x[q] is integral, this transformation must not be used.)
+*
+*  The term a[p,q] x[q] in constraint (2) can be considered as a slack
+*  variable that allows to carry bounds of column q over row p and then
+*  remove column q from the problem.
+*
+*  Constraint (2) can be written as follows:
+*
+*     sum a[p,j] x[j] = b - a[p,q] x[q].                             (3)
+*     j!=q
+*
+*  According to (1) constraint (3) is equivalent to the following
+*  inequality constraint:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],                               (4)
+*             j!=q
+*
+*  where
+*
+*            ( b - a[p,q] u[q],  if a[p,q] > 0
+*     L[p] = <                                                       (5)
+*            ( b - a[p,q] l[q],  if a[p,q] < 0
+*
+*            ( b - a[p,q] l[q],  if a[p,q] > 0
+*     U[p] = <                                                       (6)
+*            ( b - a[p,q] u[q],  if a[p,q] < 0
+*
+*  From (2) it follows that:
+*
+*              1
+*     x[q] = ------ (b - sum a[p,j] x[j]).                           (7)
+*            a[p,q]      j!=q
+*
+*  In order to eliminate x[q] from the objective row we substitute it
+*  from (6) to that row:
+*
+*     z = sum c[j] x[j] + c[q] x[q] + c[0] =
+*         j!=q
+*                                 1
+*       = sum c[j] x[j] + c[q] [------ (b - sum a[p,j] x[j])] + c0 =
+*         j!=q                  a[p,q]      j!=q
+*
+*       = sum c~[j] x[j] + c~[0],
+*         j!=q
+*                         a[p,j]                     b
+*     c~[j] = c[j] - c[q] ------,  c~0 = c0 - c[q] ------            (8)
+*                         a[p,q]                   a[p,q]
+*
+*  are values of objective coefficients and constant term, resp., in
+*  the transformed problem.
+*
+*  Note that column q is column singleton, so in the dual system of the
+*  original problem it corresponds to the following row singleton:
+*
+*     a[p,q] pi[p] + lambda[q] = c[q].                               (9)
+*
+*  In the transformed problem row (9) would be the following:
+*
+*     a[p,q] pi~[p] + lambda[q] = c~[q] = 0.                        (10)
+*
+*  Subtracting (10) from (9) we have:
+*
+*     a[p,q] (pi[p] - pi~[p]) = c[q]
+*
+*  that gives the following formula to compute multiplier for row p in
+*  solution to the original problem using its value in solution to the
+*  transformed problem:
+*
+*     pi[p] = pi~[p] + c[q] / a[p,q].                               (11)
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of column q in solution to the original problem is defined
+*  by status of row p in solution to the transformed problem and the
+*  sign of coefficient a[p,q] in the original inequality constraint (2)
+*  as follows:
+*
+*     +-----------------------+---------+--------------------+
+*     |    Status of row p    | Sign of | Status of column q |
+*     | (transformed problem) | a[p,q]  | (original problem) |
+*     +-----------------------+---------+--------------------+
+*     |        GLP_BS         |  + / -  |       GLP_BS       |
+*     |        GLP_NL         |    +    |       GLP_NU       |
+*     |        GLP_NL         |    -    |       GLP_NL       |
+*     |        GLP_NU         |    +    |       GLP_NL       |
+*     |        GLP_NU         |    -    |       GLP_NU       |
+*     |        GLP_NF         |  + / -  |       GLP_NF       |
+*     +-----------------------+---------+--------------------+
+*
+*  Value of column q is computed with formula (7). Since originally row
+*  p is equality constraint, its status is assigned GLP_NS, and value of
+*  its multiplier pi[p] is computed with formula (11).
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of column q is computed with formula (7). Row multiplier value
+*  pi[p] is computed with formula (11).
+*
+*  RECOVERING MIP SOLUTION
+*
+*  Value of column q is computed with formula (7). */
+
+struct implied_slack
+{     /* column singleton (implied slack variable) */
+      int p;
+      /* row reference number */
+      int q;
+      /* column reference number */
+      double apq;
+      /* constraint coefficient a[p,q] */
+      double b;
+      /* right-hand side of original equality constraint */
+      double c;
+      /* original objective coefficient at x[q] */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[p,j], j != q */
+};
+
+static int rcv_implied_slack(NPP *npp, void *info);
+
+void npp_implied_slack(NPP *npp, NPPCOL *q)
+{     /* process column singleton (implied slack variable) */
+      struct implied_slack *info;
+      NPPROW *p;
+      NPPAIJ *aij;
+      NPPLFE *lfe;
+      /* the column must be non-integral non-fixed singleton */
+      xassert(!q->is_int);
+      xassert(q->lb < q->ub);
+      xassert(q->ptr != NULL && q->ptr->c_next == NULL);
+      /* corresponding row must be equality constraint */
+      aij = q->ptr;
+      p = aij->row;
+      xassert(p->lb == p->ub);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_implied_slack, sizeof(struct implied_slack));
+      info->p = p->i;
+      info->q = q->j;
+      info->apq = aij->val;
+      info->b = p->lb;
+      info->c = q->coef;
+      info->ptr = NULL;
+      /* save row coefficients a[p,j], j != q, and substitute x[q]
+         into the objective row */
+      for (aij = p->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->col == q) continue; /* skip a[p,q] */
+         lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+         lfe->ref = aij->col->j;
+         lfe->val = aij->val;
+         lfe->next = info->ptr;
+         info->ptr = lfe;
+         aij->col->coef -= info->c * (aij->val / info->apq);
+      }
+      npp->c0 += info->c * (info->b / info->apq);
+      /* compute new row bounds */
+      if (info->apq > 0.0)
+      {  p->lb = (q->ub == +DBL_MAX ?
+            -DBL_MAX : info->b - info->apq * q->ub);
+         p->ub = (q->lb == -DBL_MAX ?
+            +DBL_MAX : info->b - info->apq * q->lb);
+      }
+      else
+      {  p->lb = (q->lb == -DBL_MAX ?
+            -DBL_MAX : info->b - info->apq * q->lb);
+         p->ub = (q->ub == +DBL_MAX ?
+            +DBL_MAX : info->b - info->apq * q->ub);
+      }
+      /* remove the column from the problem */
+      npp_del_col(npp, q);
+      return;
+}
+
+static int rcv_implied_slack(NPP *npp, void *_info)
+{     /* recover column singleton (implied slack variable) */
+      struct implied_slack *info = _info;
+      NPPLFE *lfe;
+      double temp;
+      if (npp->sol == GLP_SOL)
+      {  /* assign statuses to row p and column q */
+         if (npp->r_stat[info->p] == GLP_BS ||
+             npp->r_stat[info->p] == GLP_NF)
+            npp->c_stat[info->q] = npp->r_stat[info->p];
+         else if (npp->r_stat[info->p] == GLP_NL)
+            npp->c_stat[info->q] =
+               (char)(info->apq > 0.0 ? GLP_NU : GLP_NL);
+         else if (npp->r_stat[info->p] == GLP_NU)
+            npp->c_stat[info->q] =
+               (char)(info->apq > 0.0 ? GLP_NL : GLP_NU);
+         else
+         {  npp_error();
+            return 1;
+         }
+         npp->r_stat[info->p] = GLP_NS;
+      }
+      if (npp->sol != GLP_MIP)
+      {  /* compute multiplier for row p */
+         npp->r_pi[info->p] += info->c / info->apq;
+      }
+      /* compute value of column q */
+      temp = info->b;
+      for (lfe = info->ptr; lfe != NULL; lfe = lfe->next)
+         temp -= lfe->val * npp->c_value[lfe->ref];
+      npp->c_value[info->q] = temp / info->apq;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_free - process column singleton (implied free variable)
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_implied_free(NPP *npp, NPPCOL *q);
+*
+*  DESCRIPTION
+*
+*  The routine npp_implied_free processes column q:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  having non-zero coefficient in the only row p, which is inequality
+*  constraint:
+*
+*     L[p] <= sum a[p,j] x[j] + a[p,q] x[q] <= U[p],                 (2)
+*             j!=q
+*
+*  where l[q] < u[q], L[p] < U[p], L[p] > -oo and/or U[p] < +oo.
+*
+*  RETURNS
+*
+*  0 - success;
+*
+*  1 - column lower and/or upper bound(s) can be active;
+*
+*  2 - problem has no dual feasible solution.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Constraint (2) can be written as follows:
+*
+*     L[p] - sum a[p,j] x[j] <= a[p,q] x[q] <= U[p] - sum a[p,j] x[j],
+*            j!=q                                     j!=q
+*
+*  from which it follows that:
+*
+*     alfa <= a[p,q] x[q] <= beta,                                   (3)
+*
+*  where
+*
+*     alfa = inf(L[p] - sum a[p,j] x[j]) =
+*                       j!=q
+*
+*          = L[p] - sup sum a[p,j] x[j] =                            (4)
+*                       j!=q
+*
+*          = L[p] -  sum  a[p,j] u[j] -  sum  a[p,j] l[j],
+*                  j in Jp             j in Jn
+*
+*     beta = sup(L[p] - sum a[p,j] x[j]) =
+*                       j!=q
+*
+*          = L[p] - inf sum a[p,j] x[j] =                            (5)
+*                       j!=q
+*
+*          = L[p] -  sum  a[p,j] l[j] -  sum  a[p,j] u[j],
+*                  j in Jp             j in Jn
+*
+*     Jp = {j != q: a[p,j] > 0},  Jn = {j != q: a[p,j] < 0}.         (6)
+*
+*  Inequality (3) defines implied bounds of variable x[q]:
+*
+*     l'[q] <= x[q] <= u'[q],                                        (7)
+*
+*  where
+*
+*             ( alfa / a[p,q], if a[p,q] > 0
+*     l'[q] = <                                                     (8a)
+*             ( beta / a[p,q], if a[p,q] < 0
+*
+*             ( beta / a[p,q], if a[p,q] > 0
+*     u'[q] = <                                                     (8b)
+*             ( alfa / a[p,q], if a[p,q] < 0
+*
+*  Thus, if l'[q] > l[q] - eps and u'[q] < u[q] + eps, where eps is
+*  an absolute tolerance for column value, column bounds (1) cannot be
+*  active, in which case column q can be replaced by equivalent free
+*  (unbounded) column.
+*
+*  Note that column q is column singleton, so in the dual system of the
+*  original problem it corresponds to the following row singleton:
+*
+*     a[p,q] pi[p] + lambda[q] = c[q],                               (9)
+*
+*  from which it follows that:
+*
+*     pi[p] = (c[q] - lambda[q]) / a[p,q].                          (10)
+*
+*  Let x[q] be implied free (unbounded) variable. Then column q can be
+*  only basic, so its multiplier lambda[q] is equal to zero, and from
+*  (10) we have:
+*
+*     pi[p] = c[q] / a[p,q].                                        (11)
+*
+*  There are possible three cases:
+*
+*  1) pi[p] < -eps, where eps is an absolute tolerance for row
+*     multiplier. In this case, to provide dual feasibility of the
+*     original problem, row p must be active on its lower bound, and
+*     if its lower bound does not exist (L[p] = -oo), the problem has
+*     no dual feasible solution;
+*
+*  2) pi[p] > +eps. In this case row p must be active on its upper
+*     bound, and if its upper bound does not exist (U[p] = +oo), the
+*     problem has no dual feasible solution;
+*
+*  3) -eps <= pi[p] <= +eps. In this case any (either lower or upper)
+*     bound of row p can be active, because this does not affect dual
+*     feasibility.
+*
+*  Thus, in all three cases original inequality constraint (2) can be
+*  replaced by equality constraint, where the right-hand side is either
+*  lower or upper bound of row p, and bounds of column q can be removed
+*  that makes it free (unbounded). (May note that this transformation
+*  can be followed by transformation "Column singleton (implied slack
+*  variable)" performed by the routine npp_implied_slack.)
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  Status of row p in solution to the original problem is determined
+*  by its status in solution to the transformed problem and its bound,
+*  which was choosen to be active:
+*
+*     +-----------------------+--------+--------------------+
+*     |    Status of row p    | Active | Status of row p    |
+*     | (transformed problem) | bound  | (original problem) |
+*     +-----------------------+--------+--------------------+
+*     |        GLP_BS         |  L[p]  |       GLP_BS       |
+*     |        GLP_BS         |  U[p]  |       GLP_BS       |
+*     |        GLP_NS         |  L[p]  |       GLP_NL       |
+*     |        GLP_NS         |  U[p]  |       GLP_NU       |
+*     +-----------------------+--------+--------------------+
+*
+*  Value of row multiplier pi[p] (as well as value of column q) in
+*  solution to the original problem is the same as in solution to the
+*  transformed problem.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of row multiplier pi[p] in solution to the original problem is
+*  the same as in solution to the transformed problem.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct implied_free
+{     /* column singleton (implied free variable) */
+      int p;
+      /* row reference number */
+      char stat;
+      /* row status:
+         GLP_NL - active constraint on lower bound
+         GLP_NU - active constraint on upper bound */
+};
+
+static int rcv_implied_free(NPP *npp, void *info);
+
+int npp_implied_free(NPP *npp, NPPCOL *q)
+{     /* process column singleton (implied free variable) */
+      struct implied_free *info;
+      NPPROW *p;
+      NPPAIJ *apq, *aij;
+      double alfa, beta, l, u, pi, eps;
+      /* the column must be non-fixed singleton */
+      xassert(q->lb < q->ub);
+      xassert(q->ptr != NULL && q->ptr->c_next == NULL);
+      /* corresponding row must be inequality constraint */
+      apq = q->ptr;
+      p = apq->row;
+      xassert(p->lb != -DBL_MAX || p->ub != +DBL_MAX);
+      xassert(p->lb < p->ub);
+      /* compute alfa */
+      alfa = p->lb;
+      if (alfa != -DBL_MAX)
+      {  for (aij = p->ptr; aij != NULL; aij = aij->r_next)
+         {  if (aij == apq) continue; /* skip a[p,q] */
+            if (aij->val > 0.0)
+            {  if (aij->col->ub == +DBL_MAX)
+               {  alfa = -DBL_MAX;
+                  break;
+               }
+               alfa -= aij->val * aij->col->ub;
+            }
+            else /* < 0.0 */
+            {  if (aij->col->lb == -DBL_MAX)
+               {  alfa = -DBL_MAX;
+                  break;
+               }
+               alfa -= aij->val * aij->col->lb;
+            }
+         }
+      }
+      /* compute beta */
+      beta = p->ub;
+      if (beta != +DBL_MAX)
+      {  for (aij = p->ptr; aij != NULL; aij = aij->r_next)
+         {  if (aij == apq) continue; /* skip a[p,q] */
+            if (aij->val > 0.0)
+            {  if (aij->col->lb == -DBL_MAX)
+               {  beta = +DBL_MAX;
+                  break;
+               }
+               beta -= aij->val * aij->col->lb;
+            }
+            else /* < 0.0 */
+            {  if (aij->col->ub == +DBL_MAX)
+               {  beta = +DBL_MAX;
+                  break;
+               }
+               beta -= aij->val * aij->col->ub;
+            }
+         }
+      }
+      /* compute implied column lower bound l'[q] */
+      if (apq->val > 0.0)
+         l = (alfa == -DBL_MAX ? -DBL_MAX : alfa / apq->val);
+      else /* < 0.0 */
+         l = (beta == +DBL_MAX ? -DBL_MAX : beta / apq->val);
+      /* compute implied column upper bound u'[q] */
+      if (apq->val > 0.0)
+         u = (beta == +DBL_MAX ? +DBL_MAX : beta / apq->val);
+      else
+         u = (alfa == -DBL_MAX ? +DBL_MAX : alfa / apq->val);
+      /* check if column lower bound l[q] can be active */
+      if (q->lb != -DBL_MAX)
+      {  eps = 1e-9 + 1e-12 * fabs(q->lb);
+         if (l < q->lb - eps) return 1; /* yes, it can */
+      }
+      /* check if column upper bound u[q] can be active */
+      if (q->ub != +DBL_MAX)
+      {  eps = 1e-9 + 1e-12 * fabs(q->ub);
+         if (u > q->ub + eps) return 1; /* yes, it can */
+      }
+      /* okay; make column q free (unbounded) */
+      q->lb = -DBL_MAX, q->ub = +DBL_MAX;
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_implied_free, sizeof(struct implied_free));
+      info->p = p->i;
+      info->stat = -1;
+      /* compute row multiplier pi[p] */
+      pi = q->coef / apq->val;
+      /* check dual feasibility for row p */
+      if (pi > +DBL_EPSILON)
+      {  /* lower bound L[p] must be active */
+         if (p->lb != -DBL_MAX)
+nl:      {  info->stat = GLP_NL;
+            p->ub = p->lb;
+         }
+         else
+         {  if (pi > +1e-5) return 2; /* dual infeasibility */
+            /* take a chance on U[p] */
+            xassert(p->ub != +DBL_MAX);
+            goto nu;
+         }
+      }
+      else if (pi < -DBL_EPSILON)
+      {  /* upper bound U[p] must be active */
+         if (p->ub != +DBL_MAX)
+nu:      {  info->stat = GLP_NU;
+            p->lb = p->ub;
+         }
+         else
+         {  if (pi < -1e-5) return 2; /* dual infeasibility */
+            /* take a chance on L[p] */
+            xassert(p->lb != -DBL_MAX);
+            goto nl;
+         }
+      }
+      else
+      {  /* any bound (either L[p] or U[p]) can be made active  */
+         if (p->ub == +DBL_MAX)
+         {  xassert(p->lb != -DBL_MAX);
+            goto nl;
+         }
+         if (p->lb == -DBL_MAX)
+         {  xassert(p->ub != +DBL_MAX);
+            goto nu;
+         }
+         if (fabs(p->lb) <= fabs(p->ub)) goto nl; else goto nu;
+      }
+      return 0;
+}
+
+static int rcv_implied_free(NPP *npp, void *_info)
+{     /* recover column singleton (implied free variable) */
+      struct implied_free *info = _info;
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] == GLP_BS)
+            npp->r_stat[info->p] = GLP_BS;
+         else if (npp->r_stat[info->p] == GLP_NS)
+         {  xassert(info->stat == GLP_NL || info->stat == GLP_NU);
+            npp->r_stat[info->p] = info->stat;
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_eq_doublet - process row doubleton (equality constraint)
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  NPPCOL *npp_eq_doublet(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_eq_doublet processes row p, which is equality
+*  constraint having exactly two non-zero coefficients:
+*
+*     a[p,q] x[q] + a[p,r] x[r] = b.                                 (1)
+*
+*  As the result of processing one of columns q or r is eliminated from
+*  all other rows and, thus, becomes column singleton of type "implied
+*  slack variable". Row p is not changed and along with column q and r
+*  remains in the problem.
+*
+*  RETURNS
+*
+*  The routine npp_eq_doublet returns pointer to the descriptor of that
+*  column q or r which has been eliminated. If, due to some reason, the
+*  elimination was not performed, the routine returns NULL.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  First, we decide which column q or r will be eliminated. Let it be
+*  column q. Consider i-th constraint row, where column q has non-zero
+*  coefficient a[i,q] != 0:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i].                               (2)
+*              j
+*
+*  In order to eliminate column q from row (2) we subtract from it row
+*  (1) multiplied by gamma[i] = a[i,q] / a[p,q], i.e. we replace in the
+*  transformed problem row (2) by its linear combination with row (1).
+*  This transformation changes only coefficients in columns q and r,
+*  and bounds of row i as follows:
+*
+*     a~[i,q] = a[i,q] - gamma[i] a[p,q] = 0,                        (3)
+*
+*     a~[i,r] = a[i,r] - gamma[i] a[p,r],                            (4)
+*
+*       L~[i] = L[i] - gamma[i] b,                                   (5)
+*
+*       U~[i] = U[i] - gamma[i] b.                                   (6)
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  The transformation of the primal system of the original problem:
+*
+*     L <= A x <= U                                                  (7)
+*
+*  is equivalent to multiplying from the left a transformation matrix F
+*  by components of this primal system, which in the transformed problem
+*  becomes the following:
+*
+*     F L <= F A x <= F U  ==>  L~ <= A~x <= U~.                     (8)
+*
+*  The matrix F has the following structure:
+*
+*         ( 1           -gamma[1]            )
+*         (                                  )
+*         (    1        -gamma[2]            )
+*         (                                  )
+*         (      ...       ...               )
+*         (                                  )
+*     F = (          1  -gamma[p-1]          )                       (9)
+*         (                                  )
+*         (                 1                )
+*         (                                  )
+*         (             -gamma[p+1]  1       )
+*         (                                  )
+*         (                ...          ...  )
+*
+*  where its column containing elements -gamma[i] corresponds to row p
+*  of the primal system.
+*
+*  From (8) it follows that the dual system of the original problem:
+*
+*     A'pi + lambda = c,                                            (10)
+*
+*  in the transformed problem becomes the following:
+*
+*     A'F'inv(F')pi + lambda = c  ==>  (A~)'pi~ + lambda = c,       (11)
+*
+*  where:
+*
+*     pi~ = inv(F')pi                                               (12)
+*
+*  is the vector of row multipliers in the transformed problem. Thus:
+*
+*     pi = F'pi~.                                                   (13)
+*
+*  Therefore, as it follows from (13), value of multiplier for row p in
+*  solution to the original problem can be computed as follows:
+*
+*     pi[p] = pi~[p] - sum gamma[i] pi~[i],                         (14)
+*                       i
+*
+*  where pi~[i] = pi[i] is multiplier for row i (i != p).
+*
+*  Note that the statuses of all rows and columns are not changed.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Multiplier for row p in solution to the original problem is computed
+*  with formula (14).
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct eq_doublet
+{     /* row doubleton (equality constraint) */
+      int p;
+      /* row reference number */
+      double apq;
+      /* constraint coefficient a[p,q] */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[i,q], i != p */
+};
+
+static int rcv_eq_doublet(NPP *npp, void *info);
+
+NPPCOL *npp_eq_doublet(NPP *npp, NPPROW *p)
+{     /* process row doubleton (equality constraint) */
+      struct eq_doublet *info;
+      NPPROW *i;
+      NPPCOL *q, *r;
+      NPPAIJ *apq, *apr, *aiq, *air, *next;
+      NPPLFE *lfe;
+      double gamma;
+      /* the row must be doubleton equality constraint */
+      xassert(p->lb == p->ub);
+      xassert(p->ptr != NULL && p->ptr->r_next != NULL &&
+              p->ptr->r_next->r_next == NULL);
+      /* choose column to be eliminated */
+      {  NPPAIJ *a1, *a2;
+         a1 = p->ptr, a2 = a1->r_next;
+         if (fabs(a2->val) < 0.001 * fabs(a1->val))
+         {  /* only first column can be eliminated, because second one
+               has too small constraint coefficient */
+            apq = a1, apr = a2;
+         }
+         else if (fabs(a1->val) < 0.001 * fabs(a2->val))
+         {  /* only second column can be eliminated, because first one
+               has too small constraint coefficient */
+            apq = a2, apr = a1;
+         }
+         else
+         {  /* both columns are appropriate; choose that one which is
+               shorter to minimize fill-in */
+            if (npp_col_nnz(npp, a1->col) <= npp_col_nnz(npp, a2->col))
+            {  /* first column is shorter */
+               apq = a1, apr = a2;
+            }
+            else
+            {  /* second column is shorter */
+               apq = a2, apr = a1;
+            }
+         }
+      }
+      /* now columns q and r have been chosen */
+      q = apq->col, r = apr->col;
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_eq_doublet, sizeof(struct eq_doublet));
+      info->p = p->i;
+      info->apq = apq->val;
+      info->ptr = NULL;
+      /* transform each row i (i != p), where a[i,q] != 0, to eliminate
+         column q */
+      for (aiq = q->ptr; aiq != NULL; aiq = next)
+      {  next = aiq->c_next;
+         if (aiq == apq) continue; /* skip row p */
+         i = aiq->row; /* row i to be transformed */
+         /* save constraint coefficient a[i,q] */
+         if (npp->sol != GLP_MIP)
+         {  lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+            lfe->ref = i->i;
+            lfe->val = aiq->val;
+            lfe->next = info->ptr;
+            info->ptr = lfe;
+         }
+         /* find coefficient a[i,r] in row i */
+         for (air = i->ptr; air != NULL; air = air->r_next)
+            if (air->col == r) break;
+         /* if a[i,r] does not exist, create a[i,r] = 0 */
+         if (air == NULL)
+            air = npp_add_aij(npp, i, r, 0.0);
+         /* compute gamma[i] = a[i,q] / a[p,q] */
+         gamma = aiq->val / apq->val;
+         /* (row i) := (row i) - gamma[i] * (row p); see (3)-(6) */
+         /* new a[i,q] is exact zero due to elimnation; remove it from
+            row i */
+         npp_del_aij(npp, aiq);
+         /* compute new a[i,r] */
+         air->val -= gamma * apr->val;
+         /* if new a[i,r] is close to zero due to numeric cancelation,
+            remove it from row i */
+         if (fabs(air->val) <= 1e-10)
+            npp_del_aij(npp, air);
+         /* compute new lower and upper bounds of row i */
+         if (i->lb == i->ub)
+            i->lb = i->ub = (i->lb - gamma * p->lb);
+         else
+         {  if (i->lb != -DBL_MAX)
+               i->lb -= gamma * p->lb;
+            if (i->ub != +DBL_MAX)
+               i->ub -= gamma * p->lb;
+         }
+      }
+      return q;
+}
+
+static int rcv_eq_doublet(NPP *npp, void *_info)
+{     /* recover row doubleton (equality constraint) */
+      struct eq_doublet *info = _info;
+      NPPLFE *lfe;
+      double gamma, temp;
+      /* we assume that processing row p is followed by processing
+         column q as singleton of type "implied slack variable", in
+         which case row p must always be active equality constraint */
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] != GLP_NS)
+         {  npp_error();
+            return 1;
+         }
+      }
+      if (npp->sol != GLP_MIP)
+      {  /* compute value of multiplier for row p; see (14) */
+         temp = npp->r_pi[info->p];
+         for (lfe = info->ptr; lfe != NULL; lfe = lfe->next)
+         {  gamma = lfe->val / info->apq; /* a[i,q] / a[p,q] */
+            temp -= gamma * npp->r_pi[lfe->ref];
+         }
+         npp->r_pi[info->p] = temp;
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_forcing_row - process forcing row
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_forcing_row(NPP *npp, NPPROW *p, int at);
+*
+*  DESCRIPTION
+*
+*  The routine npp_forcing row processes row p of general format:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],                               (1)
+*              j
+*
+*     l[j] <= x[j] <= u[j],                                          (2)
+*
+*  where L[p] <= U[p] and l[j] < u[j] for all a[p,j] != 0. It is also
+*  assumed that:
+*
+*  1) if at = 0 then |L[p] - U'[p]| <= eps, where U'[p] is implied
+*     row upper bound (see below), eps is an absolute tolerance for row
+*     value;
+*
+*  2) if at = 1 then |U[p] - L'[p]| <= eps, where L'[p] is implied
+*     row lower bound (see below).
+*
+*  RETURNS
+*
+*  0 - success;
+*
+*  1 - cannot fix columns due to too small constraint coefficients.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Implied lower and upper bounds of row (1) are determined by bounds
+*  of corresponding columns (variables) as follows:
+*
+*     L'[p] = inf sum a[p,j] x[j] =
+*                  j
+*                                                                    (3)
+*           =  sum  a[p,j] l[j] +  sum  a[p,j] u[j],
+*            j in Jp             j in Jn
+*
+*     U'[p] = sup sum a[p,j] x[j] =
+*                                                                    (4)
+*           =  sum  a[p,j] u[j] +  sum  a[p,j] l[j],
+*            j in Jp             j in Jn
+*
+*     Jp = {j: a[p,j] > 0},  Jn = {j: a[p,j] < 0}.                   (5)
+*
+*  If L[p] =~ U'[p] (at = 0), solution can be primal feasible only when
+*  all variables take their boundary values as defined by (4):
+*
+*            ( u[j], if j in Jp
+*     x[j] = <                                                       (6)
+*            ( l[j], if j in Jn
+*
+*  Similarly, if U[p] =~ L'[p] (at = 1), solution can be primal feasible
+*  only when all variables take their boundary values as defined by (3):
+*
+*            ( l[j], if j in Jp
+*     x[j] = <                                                       (7)
+*            ( u[j], if j in Jn
+*
+*  Condition (6) or (7) allows fixing all columns (variables x[j])
+*  in row (1) on their bounds and then removing them from the problem
+*  (see the routine npp_fixed_col). Due to this row p becomes redundant,
+*  so it can be replaced by equivalent free (unbounded) row and also
+*  removed from the problem (see the routine npp_free_row).
+*
+*  1. To apply this transformation row (1) should not have coefficients
+*     whose magnitude is too small, i.e. all a[p,j] should satisfy to
+*     the following condition:
+*
+*        |a[p,j]| >= eps * max(1, |a[p,k]|),                         (8)
+*                           k
+*     where eps is a relative tolerance for constraint coefficients.
+*     Otherwise, fixing columns may be numerically unreliable and may
+*     lead to wrong solution.
+*
+*  2. The routine fixes columns and remove bounds of row p, however,
+*     it does not remove the row and columns from the problem.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  In the transformed problem row p being inactive constraint is
+*  assigned status GLP_BS (as the result of transformation of free
+*  row), and all columns in this row are assigned status GLP_NS (as the
+*  result of transformation of fixed columns).
+*
+*  Note that in the dual system of the transformed (as well as original)
+*  problem every column j in row p corresponds to the following row:
+*
+*     sum  a[i,j] pi[i] + a[p,j] pi[p] + lambda[j] = c[j],           (9)
+*     i!=p
+*
+*  from which it follows that:
+*
+*     lambda[j] = c[j] - sum a[i,j] pi[i] - a[p,j] pi[p].           (10)
+*                        i!=p
+*
+*  In the transformed problem values of all multipliers pi[i] are known
+*  (including pi[i], whose value is zero, since row p is inactive).
+*  Thus, using formula (10) it is possible to compute values of
+*  multipliers lambda[j] for all columns in row p.
+*
+*  Note also that in the original problem all columns in row p are
+*  bounded, not fixed. So status GLP_NS assigned to every such column
+*  must be changed to GLP_NL or GLP_NU depending on which bound the
+*  corresponding column has been fixed. This status change may lead to
+*  dual feasibility violation for solution of the original problem,
+*  because now column multipliers must satisfy to the following
+*  condition:
+*
+*               ( >= 0, if status of column j is GLP_NL,
+*     lambda[j] <                                                   (11)
+*               ( <= 0, if status of column j is GLP_NU.
+*
+*  If this condition holds, solution to the original problem is the
+*  same as to the transformed problem. Otherwise, we have to perform
+*  one degenerate pivoting step of the primal simplex method to obtain
+*  dual feasible (hence, optimal) solution to the original problem as
+*  follows. If, on problem transformation, row p was made active on its
+*  lower bound (case at = 0), we change its status to GLP_NL (or GLP_NS)
+*  and start increasing its multiplier pi[p]. Otherwise, if row p was
+*  made active on its upper bound (case at = 1), we change its status
+*  to GLP_NU (or GLP_NS) and start decreasing pi[p]. From (10) it
+*  follows that:
+*
+*     delta lambda[j] = - a[p,j] * delta pi[p] = - a[p,j] pi[p].    (12)
+*
+*  Simple analysis of formulae (3)-(5) shows that changing pi[p] in the
+*  specified direction causes increasing lambda[j] for every column j
+*  assigned status GLP_NL (delta lambda[j] > 0) and decreasing lambda[j]
+*  for every column j assigned status GLP_NU (delta lambda[j] < 0). It
+*  is understood that once the last lambda[q], which violates condition
+*  (11), has reached zero, multipliers lambda[j] for all columns get
+*  valid signs. Such column q can be determined as follows. Let d[j] be
+*  initial value of lambda[j] (i.e. reduced cost of column j) in the
+*  transformed problem computed with formula (10) when pi[p] = 0. Then
+*  lambda[j] = d[j] + delta lambda[j], and from (12) it follows that
+*  lambda[j] becomes zero if:
+*
+*     delta lambda[j] = - a[p,j] pi[p] = - d[j]  ==>
+*                                                                   (13)
+*     pi[p] = d[j] / a[p,j].
+*
+*  Therefore, the last column q, for which lambda[q] becomes zero, can
+*  be determined from the following condition:
+*
+*     |d[q] / a[p,q]| = max  |pi[p]| = max  |d[j] / a[p,j]|,        (14)
+*                      j in D         j in D
+*
+*  where D is a set of columns j whose, reduced costs d[j] have invalid
+*  signs, i.e. violate condition (11). (Thus, if D is empty, solution
+*  to the original problem is the same as solution to the transformed
+*  problem, and no correction is needed as was noticed above.) In
+*  solution to the original problem column q is assigned status GLP_BS,
+*  since it replaces column of auxiliary variable of row p (becoming
+*  active) in the basis, and multiplier for row p is assigned its new
+*  value, which is pi[p] = d[q] / a[p,q]. Note that due to primal
+*  degeneracy values of all columns having non-zero coefficients in row
+*  p remain unchanged.
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  Value of multiplier pi[p] in solution to the original problem is
+*  corrected in the same way as for basic solution. Values of all
+*  columns having non-zero coefficients in row p remain unchanged.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct forcing_col
+{     /* column fixed on its bound by forcing row */
+      int j;
+      /* column reference number */
+      char stat;
+      /* original column status:
+         GLP_NL - fixed on lower bound
+         GLP_NU - fixed on upper bound */
+      double a;
+      /* constraint coefficient a[p,j] */
+      double c;
+      /* objective coefficient c[j] */
+      NPPLFE *ptr;
+      /* list of non-zero coefficients a[i,j], i != p */
+      struct forcing_col *next;
+      /* pointer to another column fixed by forcing row */
+};
+
+struct forcing_row
+{     /* forcing row */
+      int p;
+      /* row reference number */
+      char stat;
+      /* status assigned to the row if it becomes active:
+         GLP_NS - active equality constraint
+         GLP_NL - inequality constraint with lower bound active
+         GLP_NU - inequality constraint with upper bound active */
+      struct forcing_col *ptr;
+      /* list of all columns having non-zero constraint coefficient
+         a[p,j] in the forcing row */
+};
+
+static int rcv_forcing_row(NPP *npp, void *info);
+
+int npp_forcing_row(NPP *npp, NPPROW *p, int at)
+{     /* process forcing row */
+      struct forcing_row *info;
+      struct forcing_col *col = NULL;
+      NPPCOL *j;
+      NPPAIJ *apj, *aij;
+      NPPLFE *lfe;
+      double big;
+      xassert(at == 0 || at == 1);
+      /* determine maximal magnitude of the row coefficients */
+      big = 1.0;
+      for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         if (big < fabs(apj->val)) big = fabs(apj->val);
+      /* if there are too small coefficients in the row, transformation
+         should not be applied */
+      for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         if (fabs(apj->val) < 1e-7 * big) return 1;
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_forcing_row, sizeof(struct forcing_row));
+      info->p = p->i;
+      if (p->lb == p->ub)
+      {  /* equality constraint */
+         info->stat = GLP_NS;
+      }
+      else if (at == 0)
+      {  /* inequality constraint; case L[p] = U'[p] */
+         info->stat = GLP_NL;
+         xassert(p->lb != -DBL_MAX);
+      }
+      else /* at == 1 */
+      {  /* inequality constraint; case U[p] = L'[p] */
+         info->stat = GLP_NU;
+         xassert(p->ub != +DBL_MAX);
+      }
+      info->ptr = NULL;
+      /* scan the forcing row, fix columns at corresponding bounds, and
+         save column information (the latter is not needed for MIP) */
+      for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+      {  /* column j has non-zero coefficient in the forcing row */
+         j = apj->col;
+         /* it must be non-fixed */
+         xassert(j->lb < j->ub);
+         /* allocate stack entry to save column information */
+         if (npp->sol != GLP_MIP)
+         {  col = dmp_get_atom(npp->stack, sizeof(struct forcing_col));
+            col->j = j->j;
+            col->stat = -1; /* will be set below */
+            col->a = apj->val;
+            col->c = j->coef;
+            col->ptr = NULL;
+            col->next = info->ptr;
+            info->ptr = col;
+         }
+         /* fix column j */
+         if (at == 0 && apj->val < 0.0 || at != 0 && apj->val > 0.0)
+         {  /* at its lower bound */
+            if (npp->sol != GLP_MIP)
+               col->stat = GLP_NL;
+            xassert(j->lb != -DBL_MAX);
+            j->ub = j->lb;
+         }
+         else
+         {  /* at its upper bound */
+            if (npp->sol != GLP_MIP)
+               col->stat = GLP_NU;
+            xassert(j->ub != +DBL_MAX);
+            j->lb = j->ub;
+         }
+         /* save column coefficients a[i,j], i != p */
+         if (npp->sol != GLP_MIP)
+         {  for (aij = j->ptr; aij != NULL; aij = aij->c_next)
+            {  if (aij == apj) continue; /* skip a[p,j] */
+               lfe = dmp_get_atom(npp->stack, sizeof(NPPLFE));
+               lfe->ref = aij->row->i;
+               lfe->val = aij->val;
+               lfe->next = col->ptr;
+               col->ptr = lfe;
+            }
+         }
+      }
+      /* make the row free (unbounded) */
+      p->lb = -DBL_MAX, p->ub = +DBL_MAX;
+      return 0;
+}
+
+static int rcv_forcing_row(NPP *npp, void *_info)
+{     /* recover forcing row */
+      struct forcing_row *info = _info;
+      struct forcing_col *col, *piv;
+      NPPLFE *lfe;
+      double d, big, temp;
+      if (npp->sol == GLP_MIP) goto done;
+      /* initially solution to the original problem is the same as
+         to the transformed problem, where row p is inactive constraint
+         with pi[p] = 0, and all columns are non-basic */
+      if (npp->sol == GLP_SOL)
+      {  if (npp->r_stat[info->p] != GLP_BS)
+         {  npp_error();
+            return 1;
+         }
+         for (col = info->ptr; col != NULL; col = col->next)
+         {  if (npp->c_stat[col->j] != GLP_NS)
+            {  npp_error();
+               return 1;
+            }
+            npp->c_stat[col->j] = col->stat; /* original status */
+         }
+      }
+      /* compute reduced costs d[j] for all columns with formula (10)
+         and store them in col.c instead objective coefficients */
+      for (col = info->ptr; col != NULL; col = col->next)
+      {  d = col->c;
+         for (lfe = col->ptr; lfe != NULL; lfe = lfe->next)
+            d -= lfe->val * npp->r_pi[lfe->ref];
+         col->c = d;
+      }
+      /* consider columns j, whose multipliers lambda[j] has wrong
+         sign in solution to the transformed problem (where lambda[j] =
+         d[j]), and choose column q, whose multipler lambda[q] reaches
+         zero last on changing row multiplier pi[p]; see (14) */
+      piv = NULL, big = 0.0;
+      for (col = info->ptr; col != NULL; col = col->next)
+      {  d = col->c; /* d[j] */
+         temp = fabs(d / col->a);
+         if (col->stat == GLP_NL)
+         {  /* column j has active lower bound */
+            if (d < 0.0 && big < temp)
+               piv = col, big = temp;
+         }
+         else if (col->stat == GLP_NU)
+         {  /* column j has active upper bound */
+            if (d > 0.0 && big < temp)
+               piv = col, big = temp;
+         }
+         else
+         {  npp_error();
+            return 1;
+         }
+      }
+      /* if column q does not exist, no correction is needed */
+      if (piv != NULL)
+      {  /* correct solution; row p becomes active constraint while
+            column q becomes basic */
+         if (npp->sol == GLP_SOL)
+         {  npp->r_stat[info->p] = info->stat;
+            npp->c_stat[piv->j] = GLP_BS;
+         }
+         /* assign new value to row multiplier pi[p] = d[p] / a[p,q] */
+         npp->r_pi[info->p] = piv->c / piv->a;
+      }
+done: return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_analyze_row - perform general row analysis
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_analyze_row(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_analyze_row performs analysis of row p of general
+*  format:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],                               (1)
+*              j
+*
+*     l[j] <= x[j] <= u[j],                                          (2)
+*
+*  where L[p] <= U[p] and l[j] <= u[j] for all a[p,j] != 0.
+*
+*  RETURNS
+*
+*  0x?0 - row lower bound does not exist or is redundant;
+*
+*  0x?1 - row lower bound can be active;
+*
+*  0x?2 - row lower bound is a forcing bound;
+*
+*  0x0? - row upper bound does not exist or is redundant;
+*
+*  0x1? - row upper bound can be active;
+*
+*  0x2? - row upper bound is a forcing bound;
+*
+*  0x33 - row bounds are inconsistent with column bounds.
+*
+*  ALGORITHM
+*
+*  Analysis of row (1) is based on analysis of its implied lower and
+*  upper bounds, which are determined by bounds of corresponding columns
+*  (variables) as follows:
+*
+*     L'[p] = inf sum a[p,j] x[j] =
+*                  j
+*                                                                    (3)
+*           =  sum  a[p,j] l[j] +  sum  a[p,j] u[j],
+*            j in Jp             j in Jn
+*
+*     U'[p] = sup sum a[p,j] x[j] =
+*                                                                    (4)
+*           =  sum  a[p,j] u[j] +  sum  a[p,j] l[j],
+*            j in Jp             j in Jn
+*
+*     Jp = {j: a[p,j] > 0},  Jn = {j: a[p,j] < 0}.                   (5)
+*
+*  (Note that bounds of all columns in row p are assumed to be correct,
+*  so L'[p] <= U'[p].)
+*
+*  Analysis of row lower bound L[p] includes the following cases:
+*
+*  1) if L[p] > U'[p] + eps, where eps is an absolute tolerance for row
+*     value, row lower bound L[p] and implied row upper bound U'[p] are
+*     inconsistent, ergo, the problem has no primal feasible solution;
+*
+*  2) if U'[p] - eps <= L[p] <= U'[p] + eps, i.e. if L[p] =~ U'[p],
+*     the row is a forcing row on its lower bound (see description of
+*     the routine npp_forcing_row);
+*
+*  3) if L[p] > L'[p] + eps, row lower bound L[p] can be active (this
+*     conclusion does not account other rows in the problem);
+*
+*  4) if L[p] <= L'[p] + eps, row lower bound L[p] cannot be active, so
+*     it is redundant and can be removed (replaced by -oo).
+*
+*  Analysis of row upper bound U[p] is performed in a similar way and
+*  includes the following cases:
+*
+*  1) if U[p] < L'[p] - eps, row upper bound U[p] and implied row lower
+*     bound L'[p] are inconsistent, ergo the problem has no primal
+*     feasible solution;
+*
+*  2) if L'[p] - eps <= U[p] <= L'[p] + eps, i.e. if U[p] =~ L'[p],
+*     the row is a forcing row on its upper bound (see description of
+*     the routine npp_forcing_row);
+*
+*  3) if U[p] < U'[p] - eps, row upper bound U[p] can be active (this
+*     conclusion does not account other rows in the problem);
+*
+*  4) if U[p] >= U'[p] - eps, row upper bound U[p] cannot be active, so
+*     it is redundant and can be removed (replaced by +oo). */
+
+int npp_analyze_row(NPP *npp, NPPROW *p)
+{     /* perform general row analysis */
+      NPPAIJ *aij;
+      int ret = 0x00;
+      double l, u, eps;
+      xassert(npp == npp);
+      /* compute implied lower bound L'[p]; see (3) */
+      l = 0.0;
+      for (aij = p->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->val > 0.0)
+         {  if (aij->col->lb == -DBL_MAX)
+            {  l = -DBL_MAX;
+               break;
+            }
+            l += aij->val * aij->col->lb;
+         }
+         else /* aij->val < 0.0 */
+         {  if (aij->col->ub == +DBL_MAX)
+            {  l = -DBL_MAX;
+               break;
+            }
+            l += aij->val * aij->col->ub;
+         }
+      }
+      /* compute implied upper bound U'[p]; see (4) */
+      u = 0.0;
+      for (aij = p->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->val > 0.0)
+         {  if (aij->col->ub == +DBL_MAX)
+            {  u = +DBL_MAX;
+               break;
+            }
+            u += aij->val * aij->col->ub;
+         }
+         else /* aij->val < 0.0 */
+         {  if (aij->col->lb == -DBL_MAX)
+            {  u = +DBL_MAX;
+               break;
+            }
+            u += aij->val * aij->col->lb;
+         }
+      }
+      /* column bounds are assumed correct, so L'[p] <= U'[p] */
+      /* check if row lower bound is consistent */
+      if (p->lb != -DBL_MAX)
+      {  eps = 1e-3 + 1e-6 * fabs(p->lb);
+         if (p->lb - eps > u)
+         {  ret = 0x33;
+            goto done;
+         }
+      }
+      /* check if row upper bound is consistent */
+      if (p->ub != +DBL_MAX)
+      {  eps = 1e-3 + 1e-6 * fabs(p->ub);
+         if (p->ub + eps < l)
+         {  ret = 0x33;
+            goto done;
+         }
+      }
+      /* check if row lower bound can be active/forcing */
+      if (p->lb != -DBL_MAX)
+      {  eps = 1e-9 + 1e-12 * fabs(p->lb);
+         if (p->lb - eps > l)
+         {  if (p->lb + eps <= u)
+               ret |= 0x01;
+            else
+               ret |= 0x02;
+         }
+      }
+      /* check if row upper bound can be active/forcing */
+      if (p->ub != +DBL_MAX)
+      {  eps = 1e-9 + 1e-12 * fabs(p->ub);
+         if (p->ub + eps < u)
+         {  /* check if the upper bound is forcing */
+            if (p->ub - eps >= l)
+               ret |= 0x10;
+            else
+               ret |= 0x20;
+         }
+      }
+done: return ret;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_inactive_bound - remove row lower/upper inactive bound
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_inactive_bound(NPP *npp, NPPROW *p, int which);
+*
+*  DESCRIPTION
+*
+*  The routine npp_inactive_bound removes lower (if which = 0) or upper
+*  (if which = 1) bound of row p:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],
+*
+*  which (bound) is assumed to be redundant.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  If which = 0, current lower bound L[p] of row p is assigned -oo.
+*  If which = 1, current upper bound U[p] of row p is assigned +oo.
+*
+*  RECOVERING BASIC SOLUTION
+*
+*  If in solution to the transformed problem row p is inactive
+*  constraint (GLP_BS), its status is not changed in solution to the
+*  original problem. Otherwise, status of row p in solution to the
+*  original problem is defined by its type before transformation and
+*  its status in solution to the transformed problem as follows:
+*
+*     +---------------------+-------+---------------+---------------+
+*     |        Row          | Flag  | Row status in | Row status in |
+*     |        type         | which | transfmd soln | original soln |
+*     +---------------------+-------+---------------+---------------+
+*     |     sum >= L[p]     |   0   |    GLP_NF     |    GLP_NL     |
+*     |     sum <= U[p]     |   1   |    GLP_NF     |    GLP_NU     |
+*     | L[p] <= sum <= U[p] |   0   |    GLP_NU     |    GLP_NU     |
+*     | L[p] <= sum <= U[p] |   1   |    GLP_NL     |    GLP_NL     |
+*     |  sum = L[p] = U[p]  |   0   |    GLP_NU     |    GLP_NS     |
+*     |  sum = L[p] = U[p]  |   1   |    GLP_NL     |    GLP_NS     |
+*     +---------------------+-------+---------------+---------------+
+*
+*  RECOVERING INTERIOR-POINT SOLUTION
+*
+*  None needed.
+*
+*  RECOVERING MIP SOLUTION
+*
+*  None needed. */
+
+struct inactive_bound
+{     /* row inactive bound */
+      int p;
+      /* row reference number */
+      char stat;
+      /* row status (if active constraint) */
+};
+
+static int rcv_inactive_bound(NPP *npp, void *info);
+
+void npp_inactive_bound(NPP *npp, NPPROW *p, int which)
+{     /* remove row lower/upper inactive bound */
+      struct inactive_bound *info;
+      if (npp->sol == GLP_SOL)
+      {  /* create transformation stack entry */
+         info = npp_push_tse(npp,
+            rcv_inactive_bound, sizeof(struct inactive_bound));
+         info->p = p->i;
+         if (p->ub == +DBL_MAX)
+            info->stat = GLP_NL;
+         else if (p->lb == -DBL_MAX)
+            info->stat = GLP_NU;
+         else if (p->lb != p->ub)
+            info->stat = (char)(which == 0 ? GLP_NU : GLP_NL);
+         else
+            info->stat = GLP_NS;
+      }
+      /* remove row inactive bound */
+      if (which == 0)
+      {  xassert(p->lb != -DBL_MAX);
+         p->lb = -DBL_MAX;
+      }
+      else if (which == 1)
+      {  xassert(p->ub != +DBL_MAX);
+         p->ub = +DBL_MAX;
+      }
+      else
+         xassert(which != which);
+      return;
+}
+
+static int rcv_inactive_bound(NPP *npp, void *_info)
+{     /* recover row status */
+      struct inactive_bound *info = _info;
+      if (npp->sol != GLP_SOL)
+      {  npp_error();
+         return 1;
+      }
+      if (npp->r_stat[info->p] == GLP_BS)
+         npp->r_stat[info->p] = GLP_BS;
+      else
+         npp->r_stat[info->p] = info->stat;
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_bounds - determine implied column bounds
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_implied_bounds(NPP *npp, NPPROW *p);
+*
+*  DESCRIPTION
+*
+*  The routine npp_implied_bounds inspects general row (constraint) p:
+*
+*     L[p] <= sum a[p,j] x[j] <= U[p],                               (1)
+*
+*     l[j] <= x[j] <= u[j],                                          (2)
+*
+*  where L[p] <= U[p] and l[j] <= u[j] for all a[p,j] != 0, to compute
+*  implied bounds of columns (variables x[j]) in this row.
+*
+*  The routine stores implied column bounds l'[j] and u'[j] in column
+*  descriptors (NPPCOL); it does not change current column bounds l[j]
+*  and u[j]. (Implied column bounds can be then used to strengthen the
+*  current column bounds; see the routines npp_implied_lower and
+*  npp_implied_upper).
+*
+*  ALGORITHM
+*
+*  Current column bounds (2) define implied lower and upper bounds of
+*  row (1) as follows:
+*
+*     L'[p] = inf sum a[p,j] x[j] =
+*                  j
+*                                                                    (3)
+*           =  sum  a[p,j] l[j] +  sum  a[p,j] u[j],
+*            j in Jp             j in Jn
+*
+*     U'[p] = sup sum a[p,j] x[j] =
+*                                                                    (4)
+*           =  sum  a[p,j] u[j] +  sum  a[p,j] l[j],
+*            j in Jp             j in Jn
+*
+*     Jp = {j: a[p,j] > 0},  Jn = {j: a[p,j] < 0}.                   (5)
+*
+*  (Note that bounds of all columns in row p are assumed to be correct,
+*  so L'[p] <= U'[p].)
+*
+*  If L[p] > L'[p] and/or U[p] < U'[p], the lower and/or upper bound of
+*  row (1) can be active, in which case such row defines implied bounds
+*  of its variables.
+*
+*  Let x[k] be some variable having in row (1) coefficient a[p,k] != 0.
+*  Consider a case when row lower bound can be active (L[p] > L'[p]):
+*
+*     sum a[p,j] x[j] >= L[p]  ==>
+*      j
+*
+*     sum a[p,j] x[j] + a[p,k] x[k] >= L[p]  ==>
+*     j!=k
+*                                                                    (6)
+*     a[p,k] x[k] >= L[p] - sum a[p,j] x[j]  ==>
+*                           j!=k
+*
+*     a[p,k] x[k] >= L[p,k],
+*
+*  where
+*
+*     L[p,k] = inf(L[p] - sum a[p,j] x[j]) =
+*                         j!=k
+*
+*            = L[p] - sup sum a[p,j] x[j] =                          (7)
+*                         j!=k
+*
+*            = L[p] - sum a[p,j] u[j] - sum a[p,j] l[j].
+*                    j in Jp\{k}       j in Jn\{k}
+*
+*  Thus:
+*
+*     x[k] >= l'[k] = L[p,k] / a[p,k],  if a[p,k] > 0,               (8)
+*
+*     x[k] <= u'[k] = L[p,k] / a[p,k],  if a[p,k] < 0.               (9)
+*
+*  where l'[k] and u'[k] are implied lower and upper bounds of variable
+*  x[k], resp.
+*
+*  Now consider a similar case when row upper bound can be active
+*  (U[p] < U'[p]):
+*
+*     sum a[p,j] x[j] <= U[p]  ==>
+*      j
+*
+*     sum a[p,j] x[j] + a[p,k] x[k] <= U[p]  ==>
+*     j!=k
+*                                                                   (10)
+*     a[p,k] x[k] <= U[p] - sum a[p,j] x[j]  ==>
+*                           j!=k
+*
+*     a[p,k] x[k] <= U[p,k],
+*
+*  where:
+*
+*     U[p,k] = sup(U[p] - sum a[p,j] x[j]) =
+*                         j!=k
+*
+*            = U[p] - inf sum a[p,j] x[j] =                         (11)
+*                         j!=k
+*
+*            = U[p] - sum a[p,j] l[j] - sum a[p,j] u[j].
+*                    j in Jp\{k}       j in Jn\{k}
+*
+*  Thus:
+*
+*     x[k] <= u'[k] = U[p,k] / a[p,k],  if a[p,k] > 0,              (12)
+*
+*     x[k] >= l'[k] = U[p,k] / a[p,k],  if a[p,k] < 0.              (13)
+*
+*  Note that in formulae (8), (9), (12), and (13) coefficient a[p,k]
+*  must not be too small in magnitude relatively to other non-zero
+*  coefficients in row (1), i.e. the following condition must hold:
+*
+*     |a[p,k]| >= eps * max(1, |a[p,j]|),                           (14)
+*                        j
+*
+*  where eps is a relative tolerance for constraint coefficients.
+*  Otherwise the implied column bounds can be numerical inreliable. For
+*  example, using formula (8) for the following inequality constraint:
+*
+*     1e-12 x1 - x2 - x3 >= 0,
+*
+*  where x1 >= -1, x2, x3, >= 0, may lead to numerically unreliable
+*  conclusion that x1 >= 0.
+*
+*  Using formulae (8), (9), (12), and (13) to compute implied bounds
+*  for one variable requires |J| operations, where J = {j: a[p,j] != 0},
+*  because this needs computing L[p,k] and U[p,k]. Thus, computing
+*  implied bounds for all variables in row (1) would require |J|^2
+*  operations, that is not a good technique. However, the total number
+*  of operations can be reduced to |J| as follows.
+*
+*  Let a[p,k] > 0. Then from (7) and (11) we have:
+*
+*     L[p,k] = L[p] - (U'[p] - a[p,k] u[k]) =
+*
+*            = L[p] - U'[p] + a[p,k] u[k],
+*
+*     U[p,k] = U[p] - (L'[p] - a[p,k] l[k]) =
+*
+*            = U[p] - L'[p] + a[p,k] l[k],
+*
+*  where L'[p] and U'[p] are implied row lower and upper bounds defined
+*  by formulae (3) and (4). Substituting these expressions into (8) and
+*  (12) gives:
+*
+*     l'[k] = L[p,k] / a[p,k] = u[k] + (L[p] - U'[p]) / a[p,k],     (15)
+*
+*     u'[k] = U[p,k] / a[p,k] = l[k] + (U[p] - L'[p]) / a[p,k].     (16)
+*
+*  Similarly, if a[p,k] < 0, according to (7) and (11) we have:
+*
+*     L[p,k] = L[p] - (U'[p] - a[p,k] l[k]) =
+*
+*            = L[p] - U'[p] + a[p,k] l[k],
+*
+*     U[p,k] = U[p] - (L'[p] - a[p,k] u[k]) =
+*
+*            = U[p] - L'[p] + a[p,k] u[k],
+*
+*  and substituting these expressions into (8) and (12) gives:
+*
+*     l'[k] = U[p,k] / a[p,k] = u[k] + (U[p] - L'[p]) / a[p,k],     (17)
+*
+*     u'[k] = L[p,k] / a[p,k] = l[k] + (L[p] - U'[p]) / a[p,k].     (18)
+*
+*  Note that formulae (15)-(18) can be used only if L'[p] and U'[p]
+*  exist. However, if for some variable x[j] it happens that l[j] = -oo
+*  and/or u[j] = +oo, values of L'[p] (if a[p,j] > 0) and/or U'[p] (if
+*  a[p,j] < 0) are undefined. Consider, therefore, the most general
+*  situation, when some column bounds (2) may not exist.
+*
+*  Let:
+*
+*     J' = {j : (a[p,j] > 0 and l[j] = -oo) or
+*                                                                   (19)
+*               (a[p,j] < 0 and u[j] = +oo)}.
+*
+*  Then (assuming that row upper bound U[p] can be active) the following
+*  three cases are possible:
+*
+*  1) |J'| = 0. In this case L'[p] exists, thus, for all variables x[j]
+*     in row (1) we can use formulae (16) and (17);
+*
+*  2) J' = {k}. In this case L'[p] = -oo, however, U[p,k] (11) exists,
+*     so for variable x[k] we can use formulae (12) and (13). Note that
+*     for all other variables x[j] (j != k) l'[j] = -oo (if a[p,j] < 0)
+*     or u'[j] = +oo (if a[p,j] > 0);
+*
+*  3) |J'| > 1. In this case for all variables x[j] in row [1] we have
+*     l'[j] = -oo (if a[p,j] < 0) or u'[j] = +oo (if a[p,j] > 0).
+*
+*  Similarly, let:
+*
+*     J'' = {j : (a[p,j] > 0 and u[j] = +oo) or
+*                                                                   (20)
+*                (a[p,j] < 0 and l[j] = -oo)}.
+*
+*  Then (assuming that row lower bound L[p] can be active) the following
+*  three cases are possible:
+*
+*  1) |J''| = 0. In this case U'[p] exists, thus, for all variables x[j]
+*     in row (1) we can use formulae (15) and (18);
+*
+*  2) J'' = {k}. In this case U'[p] = +oo, however, L[p,k] (7) exists,
+*     so for variable x[k] we can use formulae (8) and (9). Note that
+*     for all other variables x[j] (j != k) l'[j] = -oo (if a[p,j] > 0)
+*     or u'[j] = +oo (if a[p,j] < 0);
+*
+*  3) |J''| > 1. In this case for all variables x[j] in row (1) we have
+*     l'[j] = -oo (if a[p,j] > 0) or u'[j] = +oo (if a[p,j] < 0). */
+
+void npp_implied_bounds(NPP *npp, NPPROW *p)
+{     NPPAIJ *apj, *apk;
+      double big, eps, temp;
+      xassert(npp == npp);
+      /* initialize implied bounds for all variables and determine
+         maximal magnitude of row coefficients a[p,j] */
+      big = 1.0;
+      for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+      {  apj->col->ll.ll = -DBL_MAX, apj->col->uu.uu = +DBL_MAX;
+         if (big < fabs(apj->val)) big = fabs(apj->val);
+      }
+      eps = 1e-6 * big;
+      /* process row lower bound (assuming that it can be active) */
+      if (p->lb != -DBL_MAX)
+      {  apk = NULL;
+         for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         {  if (apj->val > 0.0 && apj->col->ub == +DBL_MAX ||
+                apj->val < 0.0 && apj->col->lb == -DBL_MAX)
+            {  if (apk == NULL)
+                  apk = apj;
+               else
+                  goto skip1;
+            }
+         }
+         /* if a[p,k] = NULL then |J'| = 0 else J' = { k } */
+         temp = p->lb;
+         for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         {  if (apj == apk)
+               /* skip a[p,k] */;
+            else if (apj->val > 0.0)
+               temp -= apj->val * apj->col->ub;
+            else /* apj->val < 0.0 */
+               temp -= apj->val * apj->col->lb;
+         }
+         /* compute column implied bounds */
+         if (apk == NULL)
+         {  /* temp = L[p] - U'[p] */
+            for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+            {  if (apj->val >= +eps)
+               {  /* l'[j] := u[j] + (L[p] - U'[p]) / a[p,j] */
+                  apj->col->ll.ll = apj->col->ub + temp / apj->val;
+               }
+               else if (apj->val <= -eps)
+               {  /* u'[j] := l[j] + (L[p] - U'[p]) / a[p,j] */
+                  apj->col->uu.uu = apj->col->lb + temp / apj->val;
+               }
+            }
+         }
+         else
+         {  /* temp = L[p,k] */
+            if (apk->val >= +eps)
+            {  /* l'[k] := L[p,k] / a[p,k] */
+               apk->col->ll.ll = temp / apk->val;
+            }
+            else if (apk->val <= -eps)
+            {  /* u'[k] := L[p,k] / a[p,k] */
+               apk->col->uu.uu = temp / apk->val;
+            }
+         }
+skip1:   ;
+      }
+      /* process row upper bound (assuming that it can be active) */
+      if (p->ub != +DBL_MAX)
+      {  apk = NULL;
+         for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         {  if (apj->val > 0.0 && apj->col->lb == -DBL_MAX ||
+                apj->val < 0.0 && apj->col->ub == +DBL_MAX)
+            {  if (apk == NULL)
+                  apk = apj;
+               else
+                  goto skip2;
+            }
+         }
+         /* if a[p,k] = NULL then |J''| = 0 else J'' = { k } */
+         temp = p->ub;
+         for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+         {  if (apj == apk)
+               /* skip a[p,k] */;
+            else if (apj->val > 0.0)
+               temp -= apj->val * apj->col->lb;
+            else /* apj->val < 0.0 */
+               temp -= apj->val * apj->col->ub;
+         }
+         /* compute column implied bounds */
+         if (apk == NULL)
+         {  /* temp = U[p] - L'[p] */
+            for (apj = p->ptr; apj != NULL; apj = apj->r_next)
+            {  if (apj->val >= +eps)
+               {  /* u'[j] := l[j] + (U[p] - L'[p]) / a[p,j] */
+                  apj->col->uu.uu = apj->col->lb + temp / apj->val;
+               }
+               else if (apj->val <= -eps)
+               {  /* l'[j] := u[j] + (U[p] - L'[p]) / a[p,j] */
+                  apj->col->ll.ll = apj->col->ub + temp / apj->val;
+               }
+            }
+         }
+         else
+         {  /* temp = U[p,k] */
+            if (apk->val >= +eps)
+            {  /* u'[k] := U[p,k] / a[p,k] */
+               apk->col->uu.uu = temp / apk->val;
+            }
+            else if (apk->val <= -eps)
+            {  /* l'[k] := U[p,k] / a[p,k] */
+               apk->col->ll.ll = temp / apk->val;
+            }
+         }
+skip2:   ;
+      }
+      return;
+}
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp4.c b/test/monniaux/glpk-4.65/src/npp/npp4.c
new file mode 100644
index 00000000..d7dd0e86
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp4.c
@@ -0,0 +1,1414 @@
+/* npp4.c */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+/***********************************************************************
+*  NAME
+*
+*  npp_binarize_prob - binarize MIP problem
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_binarize_prob(NPP *npp);
+*
+*  DESCRIPTION
+*
+*  The routine npp_binarize_prob replaces in the original MIP problem
+*  every integer variable:
+*
+*     l[q] <= x[q] <= u[q],                                          (1)
+*
+*  where l[q] < u[q], by an equivalent sum of binary variables.
+*
+*  RETURNS
+*
+*  The routine returns the number of integer variables for which the
+*  transformation failed, because u[q] - l[q] > d_max.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  If variable x[q] has non-zero lower bound, it is first processed
+*  with the routine npp_lbnd_col. Thus, we can assume that:
+*
+*     0 <= x[q] <= u[q].                                             (2)
+*
+*  If u[q] = 1, variable x[q] is already binary, so further processing
+*  is not needed. Let, therefore, that 2 <= u[q] <= d_max, and n be a
+*  smallest integer such that u[q] <= 2^n - 1 (n >= 2, since u[q] >= 2).
+*  Then variable x[q] can be replaced by the following sum:
+*
+*            n-1
+*     x[q] = sum 2^k x[k],                                           (3)
+*            k=0
+*
+*  where x[k] are binary columns (variables). If u[q] < 2^n - 1, the
+*  following additional inequality constraint must be also included in
+*  the transformed problem:
+*
+*     n-1
+*     sum 2^k x[k] <= u[q].                                          (4)
+*     k=0
+*
+*  Note: Assuming that in the transformed problem x[q] becomes binary
+*  variable x[0], this transformation causes new n-1 binary variables
+*  to appear.
+*
+*  Substituting x[q] from (3) to the objective row gives:
+*
+*     z = sum c[j] x[j] + c[0] =
+*          j
+*
+*       = sum c[j] x[j] + c[q] x[q] + c[0] =
+*         j!=q
+*                              n-1
+*       = sum c[j] x[j] + c[q] sum 2^k x[k] + c[0] =
+*         j!=q                 k=0
+*                         n-1
+*       = sum c[j] x[j] + sum c[k] x[k] + c[0],
+*         j!=q            k=0
+*
+*  where:
+*
+*     c[k] = 2^k c[q],  k = 0, ..., n-1.                             (5)
+*
+*  And substituting x[q] from (3) to i-th constraint row i gives:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i]  ==>
+*              j
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] x[q] <= U[i]  ==>
+*             j!=q
+*                                      n-1
+*     L[i] <= sum a[i,j] x[j] + a[i,q] sum 2^k x[k] <= U[i]  ==>
+*             j!=q                     k=0
+*                               n-1
+*     L[i] <= sum a[i,j] x[j] + sum a[i,k] x[k] <= U[i],
+*             j!=q              k=0
+*
+*  where:
+*
+*     a[i,k] = 2^k a[i,q],  k = 0, ..., n-1.                         (6)
+*
+*  RECOVERING SOLUTION
+*
+*  Value of variable x[q] is computed with formula (3). */
+
+struct binarize
+{     int q;
+      /* column reference number for x[q] = x[0] */
+      int j;
+      /* column reference number for x[1]; x[2] has reference number
+         j+1, x[3] - j+2, etc. */
+      int n;
+      /* total number of binary variables, n >= 2 */
+};
+
+static int rcv_binarize_prob(NPP *npp, void *info);
+
+int npp_binarize_prob(NPP *npp)
+{     /* binarize MIP problem */
+      struct binarize *info;
+      NPPROW *row;
+      NPPCOL *col, *bin;
+      NPPAIJ *aij;
+      int u, n, k, temp, nfails, nvars, nbins, nrows;
+      /* new variables will be added to the end of the column list, so
+         we go from the end to beginning of the column list */
+      nfails = nvars = nbins = nrows = 0;
+      for (col = npp->c_tail; col != NULL; col = col->prev)
+      {  /* skip continuous variable */
+         if (!col->is_int) continue;
+         /* skip fixed variable */
+         if (col->lb == col->ub) continue;
+         /* skip binary variable */
+         if (col->lb == 0.0 && col->ub == 1.0) continue;
+         /* check if the transformation is applicable */
+         if (col->lb < -1e6 || col->ub > +1e6 ||
+             col->ub - col->lb > 4095.0)
+         {  /* unfortunately, not */
+            nfails++;
+            continue;
+         }
+         /* process integer non-binary variable x[q] */
+         nvars++;
+         /* make x[q] non-negative, if its lower bound is non-zero */
+         if (col->lb != 0.0)
+            npp_lbnd_col(npp, col);
+         /* now 0 <= x[q] <= u[q] */
+         xassert(col->lb == 0.0);
+         u = (int)col->ub;
+         xassert(col->ub == (double)u);
+         /* if x[q] is binary, further processing is not needed */
+         if (u == 1) continue;
+         /* determine smallest n such that u <= 2^n - 1 (thus, n is the
+            number of binary variables needed) */
+         n = 2, temp = 4;
+         while (u >= temp)
+            n++, temp += temp;
+         nbins += n;
+         /* create transformation stack entry */
+         info = npp_push_tse(npp,
+            rcv_binarize_prob, sizeof(struct binarize));
+         info->q = col->j;
+         info->j = 0; /* will be set below */
+         info->n = n;
+         /* if u < 2^n - 1, we need one additional row for (4) */
+         if (u < temp - 1)
+         {  row = npp_add_row(npp), nrows++;
+            row->lb = -DBL_MAX, row->ub = u;
+         }
+         else
+            row = NULL;
+         /* in the transformed problem variable x[q] becomes binary
+            variable x[0], so its objective and constraint coefficients
+            are not changed */
+         col->ub = 1.0;
+         /* include x[0] into constraint (4) */
+         if (row != NULL)
+            npp_add_aij(npp, row, col, 1.0);
+         /* add other binary variables x[1], ..., x[n-1] */
+         for (k = 1, temp = 2; k < n; k++, temp += temp)
+         {  /* add new binary variable x[k] */
+            bin = npp_add_col(npp);
+            bin->is_int = 1;
+            bin->lb = 0.0, bin->ub = 1.0;
+            bin->coef = (double)temp * col->coef;
+            /* store column reference number for x[1] */
+            if (info->j == 0)
+               info->j = bin->j;
+            else
+               xassert(info->j + (k-1) == bin->j);
+            /* duplicate constraint coefficients for x[k]; this also
+               automatically includes x[k] into constraint (4) */
+            for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+               npp_add_aij(npp, aij->row, bin, (double)temp * aij->val);
+         }
+      }
+      if (nvars > 0)
+         xprintf("%d integer variable(s) were replaced by %d binary one"
+            "s\n", nvars, nbins);
+      if (nrows > 0)
+         xprintf("%d row(s) were added due to binarization\n", nrows);
+      if (nfails > 0)
+         xprintf("Binarization failed for %d integer variable(s)\n",
+            nfails);
+      return nfails;
+}
+
+static int rcv_binarize_prob(NPP *npp, void *_info)
+{     /* recovery binarized variable */
+      struct binarize *info = _info;
+      int k, temp;
+      double sum;
+      /* compute value of x[q]; see formula (3) */
+      sum = npp->c_value[info->q];
+      for (k = 1, temp = 2; k < info->n; k++, temp += temp)
+         sum += (double)temp * npp->c_value[info->j + (k-1)];
+      npp->c_value[info->q] = sum;
+      return 0;
+}
+
+/**********************************************************************/
+
+struct elem
+{     /* linear form element a[j] x[j] */
+      double aj;
+      /* non-zero coefficient value */
+      NPPCOL *xj;
+      /* pointer to variable (column) */
+      struct elem *next;
+      /* pointer to another term */
+};
+
+static struct elem *copy_form(NPP *npp, NPPROW *row, double s)
+{     /* copy linear form */
+      NPPAIJ *aij;
+      struct elem *ptr, *e;
+      ptr = NULL;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  e = dmp_get_atom(npp->pool, sizeof(struct elem));
+         e->aj = s * aij->val;
+         e->xj = aij->col;
+         e->next = ptr;
+         ptr = e;
+      }
+      return ptr;
+}
+
+static void drop_form(NPP *npp, struct elem *ptr)
+{     /* drop linear form */
+      struct elem *e;
+      while (ptr != NULL)
+      {  e = ptr;
+         ptr = e->next;
+         dmp_free_atom(npp->pool, e, sizeof(struct elem));
+      }
+      return;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_is_packing - test if constraint is packing inequality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_is_packing(NPP *npp, NPPROW *row);
+*
+*  RETURNS
+*
+*  If the specified row (constraint) is packing inequality (see below),
+*  the routine npp_is_packing returns non-zero. Otherwise, it returns
+*  zero.
+*
+*  PACKING INEQUALITIES
+*
+*  In canonical format the packing inequality is the following:
+*
+*     sum  x[j] <= 1,                                                (1)
+*    j in J
+*
+*  where all variables x[j] are binary. This inequality expresses the
+*  condition that in any integer feasible solution at most one variable
+*  from set J can take non-zero (unity) value while other variables
+*  must be equal to zero. W.l.o.g. it is assumed that |J| >= 2, because
+*  if J is empty or |J| = 1, the inequality (1) is redundant.
+*
+*  In general case the packing inequality may include original variables
+*  x[j] as well as their complements x~[j]:
+*
+*     sum   x[j] + sum   x~[j] <= 1,                                 (2)
+*    j in Jp      j in Jn
+*
+*  where Jp and Jn are not intersected. Therefore, using substitution
+*  x~[j] = 1 - x[j] gives the packing inequality in generalized format:
+*
+*     sum   x[j] - sum   x[j] <= 1 - |Jn|.                           (3)
+*    j in Jp      j in Jn */
+
+int npp_is_packing(NPP *npp, NPPROW *row)
+{     /* test if constraint is packing inequality */
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int b;
+      xassert(npp == npp);
+      if (!(row->lb == -DBL_MAX && row->ub != +DBL_MAX))
+         return 0;
+      b = 1;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  col = aij->col;
+         if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+            return 0;
+         if (aij->val == +1.0)
+            ;
+         else if (aij->val == -1.0)
+            b--;
+         else
+            return 0;
+      }
+      if (row->ub != (double)b) return 0;
+      return 1;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_hidden_packing - identify hidden packing inequality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_hidden_packing(NPP *npp, NPPROW *row);
+*
+*  DESCRIPTION
+*
+*  The routine npp_hidden_packing processes specified inequality
+*  constraint, which includes only binary variables, and the number of
+*  the variables is not less than two. If the original inequality is
+*  equivalent to a packing inequality, the routine replaces it by this
+*  equivalent inequality. If the original constraint is double-sided
+*  inequality, it is replaced by a pair of single-sided inequalities,
+*  if necessary.
+*
+*  RETURNS
+*
+*  If the original inequality constraint was replaced by equivalent
+*  packing inequality, the routine npp_hidden_packing returns non-zero.
+*  Otherwise, it returns zero.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Consider an inequality constraint:
+*
+*     sum  a[j] x[j] <= b,                                           (1)
+*    j in J
+*
+*  where all variables x[j] are binary, and |J| >= 2. (In case of '>='
+*  inequality it can be transformed to '<=' format by multiplying both
+*  its sides by -1.)
+*
+*  Let Jp = {j: a[j] > 0}, Jn = {j: a[j] < 0}. Performing substitution
+*  x[j] = 1 - x~[j] for all j in Jn, we have:
+*
+*     sum   a[j] x[j] <= b  ==>
+*    j in J
+*
+*     sum   a[j] x[j] + sum   a[j] x[j] <= b  ==>
+*    j in Jp           j in Jn
+*
+*     sum   a[j] x[j] + sum   a[j] (1 - x~[j]) <= b  ==>
+*    j in Jp           j in Jn
+*
+*     sum   a[j] x[j] - sum   a[j] x~[j] <= b - sum   a[j].
+*    j in Jp           j in Jn                 j in Jn
+*
+*  Thus, meaning the transformation above, we can assume that in
+*  inequality (1) all coefficients a[j] are positive. Moreover, we can
+*  assume that a[j] <= b. In fact, let a[j] > b; then the following
+*  three cases are possible:
+*
+*  1) b < 0. In this case inequality (1) is infeasible, so the problem
+*     has no feasible solution (see the routine npp_analyze_row);
+*
+*  2) b = 0. In this case inequality (1) is a forcing inequality on its
+*     upper bound (see the routine npp_forcing row), from which it
+*     follows that all variables x[j] should be fixed at zero;
+*
+*  3) b > 0. In this case inequality (1) defines an implied zero upper
+*     bound for variable x[j] (see the routine npp_implied_bounds), from
+*     which it follows that x[j] should be fixed at zero.
+*
+*  It is assumed that all three cases listed above have been recognized
+*  by the routine npp_process_prob, which performs basic MIP processing
+*  prior to a call the routine npp_hidden_packing. So, if one of these
+*  cases occurs, we should just skip processing such constraint.
+*
+*  Thus, let 0 < a[j] <= b. Then it is obvious that constraint (1) is
+*  equivalent to packing inquality only if:
+*
+*     a[j] + a[k] > b + eps                                          (2)
+*
+*  for all j, k in J, j != k, where eps is an absolute tolerance for
+*  row (linear form) value. Checking the condition (2) for all j and k,
+*  j != k, requires time O(|J|^2). However, this time can be reduced to
+*  O(|J|), if use minimal a[j] and a[k], in which case it is sufficient
+*  to check the condition (2) only once.
+*
+*  Once the original inequality (1) is replaced by equivalent packing
+*  inequality, we need to perform back substitution x~[j] = 1 - x[j] for
+*  all j in Jn (see above).
+*
+*  RECOVERING SOLUTION
+*
+*  None needed. */
+
+static int hidden_packing(NPP *npp, struct elem *ptr, double *_b)
+{     /* process inequality constraint: sum a[j] x[j] <= b;
+         0 - specified row is NOT hidden packing inequality;
+         1 - specified row is packing inequality;
+         2 - specified row is hidden packing inequality. */
+      struct elem *e, *ej, *ek;
+      int neg;
+      double b = *_b, eps;
+      xassert(npp == npp);
+      /* a[j] must be non-zero, x[j] must be binary, for all j in J */
+      for (e = ptr; e != NULL; e = e->next)
+      {  xassert(e->aj != 0.0);
+         xassert(e->xj->is_int);
+         xassert(e->xj->lb == 0.0 && e->xj->ub == 1.0);
+      }
+      /* check if the specified inequality constraint already has the
+         form of packing inequality */
+      neg = 0; /* neg is |Jn| */
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (e->aj == +1.0)
+            ;
+         else if (e->aj == -1.0)
+            neg++;
+         else
+            break;
+      }
+      if (e == NULL)
+      {  /* all coefficients a[j] are +1 or -1; check rhs b */
+         if (b == (double)(1 - neg))
+         {  /* it is packing inequality; no processing is needed */
+            return 1;
+         }
+      }
+      /* substitute x[j] = 1 - x~[j] for all j in Jn to make all a[j]
+         positive; the result is a~[j] = |a[j]| and new rhs b */
+      for (e = ptr; e != NULL; e = e->next)
+         if (e->aj < 0) b -= e->aj;
+      /* now a[j] > 0 for all j in J (actually |a[j]| are used) */
+      /* if a[j] > b, skip processing--this case must not appear */
+      for (e = ptr; e != NULL; e = e->next)
+         if (fabs(e->aj) > b) return 0;
+      /* now 0 < a[j] <= b for all j in J */
+      /* find two minimal coefficients a[j] and a[k], j != k */
+      ej = NULL;
+      for (e = ptr; e != NULL; e = e->next)
+         if (ej == NULL || fabs(ej->aj) > fabs(e->aj)) ej = e;
+      xassert(ej != NULL);
+      ek = NULL;
+      for (e = ptr; e != NULL; e = e->next)
+         if (e != ej)
+            if (ek == NULL || fabs(ek->aj) > fabs(e->aj)) ek = e;
+      xassert(ek != NULL);
+      /* the specified constraint is equivalent to packing inequality
+         iff a[j] + a[k] > b + eps */
+      eps = 1e-3 + 1e-6 * fabs(b);
+      if (fabs(ej->aj) + fabs(ek->aj) <= b + eps) return 0;
+      /* perform back substitution x~[j] = 1 - x[j] and construct the
+         final equivalent packing inequality in generalized format */
+      b = 1.0;
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (e->aj > 0.0)
+            e->aj = +1.0;
+         else /* e->aj < 0.0 */
+            e->aj = -1.0, b -= 1.0;
+      }
+      *_b = b;
+      return 2;
+}
+
+int npp_hidden_packing(NPP *npp, NPPROW *row)
+{     /* identify hidden packing inequality */
+      NPPROW *copy;
+      NPPAIJ *aij;
+      struct elem *ptr, *e;
+      int kase, ret, count = 0;
+      double b;
+      /* the row must be inequality constraint */
+      xassert(row->lb < row->ub);
+      for (kase = 0; kase <= 1; kase++)
+      {  if (kase == 0)
+         {  /* process row upper bound */
+            if (row->ub == +DBL_MAX) continue;
+            ptr = copy_form(npp, row, +1.0);
+            b = + row->ub;
+         }
+         else
+         {  /* process row lower bound */
+            if (row->lb == -DBL_MAX) continue;
+            ptr = copy_form(npp, row, -1.0);
+            b = - row->lb;
+         }
+         /* now the inequality has the form "sum a[j] x[j] <= b" */
+         ret = hidden_packing(npp, ptr, &b);
+         xassert(0 <= ret && ret <= 2);
+         if (kase == 1 && ret == 1 || ret == 2)
+         {  /* the original inequality has been identified as hidden
+               packing inequality */
+            count++;
+#ifdef GLP_DEBUG
+            xprintf("Original constraint:\n");
+            for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+               xprintf(" %+g x%d", aij->val, aij->col->j);
+            if (row->lb != -DBL_MAX) xprintf(", >= %g", row->lb);
+            if (row->ub != +DBL_MAX) xprintf(", <= %g", row->ub);
+            xprintf("\n");
+            xprintf("Equivalent packing inequality:\n");
+            for (e = ptr; e != NULL; e = e->next)
+               xprintf(" %sx%d", e->aj > 0.0 ? "+" : "-", e->xj->j);
+            xprintf(", <= %g\n", b);
+#endif
+            if (row->lb == -DBL_MAX || row->ub == +DBL_MAX)
+            {  /* the original row is single-sided inequality; no copy
+                  is needed */
+               copy = NULL;
+            }
+            else
+            {  /* the original row is double-sided inequality; we need
+                  to create its copy for other bound before replacing it
+                  with the equivalent inequality */
+               copy = npp_add_row(npp);
+               if (kase == 0)
+               {  /* the copy is for lower bound */
+                  copy->lb = row->lb, copy->ub = +DBL_MAX;
+               }
+               else
+               {  /* the copy is for upper bound */
+                  copy->lb = -DBL_MAX, copy->ub = row->ub;
+               }
+               /* copy original row coefficients */
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  npp_add_aij(npp, copy, aij->col, aij->val);
+            }
+            /* replace the original inequality by equivalent one */
+            npp_erase_row(npp, row);
+            row->lb = -DBL_MAX, row->ub = b;
+            for (e = ptr; e != NULL; e = e->next)
+               npp_add_aij(npp, row, e->xj, e->aj);
+            /* continue processing lower bound for the copy */
+            if (copy != NULL) row = copy;
+         }
+         drop_form(npp, ptr);
+      }
+      return count;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_implied_packing - identify implied packing inequality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_implied_packing(NPP *npp, NPPROW *row, int which,
+*     NPPCOL *var[], char set[]);
+*
+*  DESCRIPTION
+*
+*  The routine npp_implied_packing processes specified row (constraint)
+*  of general format:
+*
+*     L <= sum a[j] x[j] <= U.                                       (1)
+*           j
+*
+*  If which = 0, only lower bound L, which must exist, is considered,
+*  while upper bound U is ignored. Similarly, if which = 1, only upper
+*  bound U, which must exist, is considered, while lower bound L is
+*  ignored. Thus, if the specified row is a double-sided inequality or
+*  equality constraint, this routine should be called twice for both
+*  lower and upper bounds.
+*
+*  The routine npp_implied_packing attempts to find a non-trivial (i.e.
+*  having not less than two binary variables) packing inequality:
+*
+*     sum   x[j] - sum   x[j] <= 1 - |Jn|,                           (2)
+*    j in Jp      j in Jn
+*
+*  which is relaxation of the constraint (1) in the sense that any
+*  solution satisfying to that constraint also satisfies to the packing
+*  inequality (2). If such relaxation exists, the routine stores
+*  pointers to descriptors of corresponding binary variables and their
+*  flags, resp., to locations var[1], var[2], ..., var[len] and set[1],
+*  set[2], ..., set[len], where set[j] = 0 means that j in Jp and
+*  set[j] = 1 means that j in Jn.
+*
+*  RETURNS
+*
+*  The routine npp_implied_packing returns len, which is the total
+*  number of binary variables in the packing inequality found, len >= 2.
+*  However, if the relaxation does not exist, the routine returns zero.
+*
+*  ALGORITHM
+*
+*  If which = 0, the constraint coefficients (1) are multiplied by -1
+*  and b is assigned -L; if which = 1, the constraint coefficients (1)
+*  are not changed and b is assigned +U. In both cases the specified
+*  constraint gets the following format:
+*
+*     sum a[j] x[j] <= b.                                            (3)
+*      j
+*
+*  (Note that (3) is a relaxation of (1), because one of bounds L or U
+*  is ignored.)
+*
+*  Let J be set of binary variables, Kp be set of non-binary (integer
+*  or continuous) variables with a[j] > 0, and Kn be set of non-binary
+*  variables with a[j] < 0. Then the inequality (3) can be written as
+*  follows:
+*
+*     sum  a[j] x[j] <= b - sum   a[j] x[j] - sum   a[j] x[j].       (4)
+*    j in J                j in Kp           j in Kn
+*
+*  To get rid of non-binary variables we can replace the inequality (4)
+*  by the following relaxed inequality:
+*
+*     sum  a[j] x[j] <= b~,                                          (5)
+*    j in J
+*
+*  where:
+*
+*     b~ = sup(b - sum   a[j] x[j] - sum   a[j] x[j]) =
+*                 j in Kp           j in Kn
+*
+*        = b - inf sum   a[j] x[j] - inf sum   a[j] x[j] =           (6)
+*                 j in Kp               j in Kn
+*
+*        = b - sum   a[j] l[j] - sum   a[j] u[j].
+*             j in Kp           j in Kn
+*
+*  Note that if lower bound l[j] (if j in Kp) or upper bound u[j]
+*  (if j in Kn) of some non-binary variable x[j] does not exist, then
+*  formally b = +oo, in which case further analysis is not performed.
+*
+*  Let Bp = {j in J: a[j] > 0}, Bn = {j in J: a[j] < 0}. To make all
+*  the inequality coefficients in (5) positive, we replace all x[j] in
+*  Bn by their complementaries, substituting x[j] = 1 - x~[j] for all
+*  j in Bn, that gives:
+*
+*     sum   a[j] x[j] - sum   a[j] x~[j] <= b~ - sum   a[j].         (7)
+*    j in Bp           j in Bn                  j in Bn
+*
+*  This inequality is a relaxation of the original constraint (1), and
+*  it is a binary knapsack inequality. Writing it in the standard format
+*  we have:
+*
+*     sum  alfa[j] z[j] <= beta,                                     (8)
+*    j in J
+*
+*  where:
+*               ( + a[j],   if j in Bp,
+*     alfa[j] = <                                                    (9)
+*               ( - a[j],   if j in Bn,
+*
+*               ( x[j],     if j in Bp,
+*        z[j] = <                                                   (10)
+*               ( 1 - x[j], if j in Bn,
+*
+*        beta = b~ - sum   a[j].                                    (11)
+*                   j in Bn
+*
+*  In the inequality (8) all coefficients are positive, therefore, the
+*  packing relaxation to be found for this inequality is the following:
+*
+*     sum  z[j] <= 1.                                               (12)
+*    j in P
+*
+*  It is obvious that set P within J, which we would like to find, must
+*  satisfy to the following condition:
+*
+*     alfa[j] + alfa[k] > beta + eps  for all j, k in P, j != k,    (13)
+*
+*  where eps is an absolute tolerance for value of the linear form.
+*  Thus, it is natural to take P = {j: alpha[j] > (beta + eps) / 2}.
+*  Moreover, if in the equality (8) there exist coefficients alfa[k],
+*  for which alfa[k] <= (beta + eps) / 2, but which, nevertheless,
+*  satisfies to the condition (13) for all j in P, *one* corresponding
+*  variable z[k] (having, for example, maximal coefficient alfa[k]) can
+*  be included in set P, that allows increasing the number of binary
+*  variables in (12) by one.
+*
+*  Once the set P has been built, for the inequality (12) we need to
+*  perform back substitution according to (10) in order to express it
+*  through the original binary variables. As the result of such back
+*  substitution the relaxed packing inequality get its final format (2),
+*  where Jp = J intersect Bp, and Jn = J intersect Bn. */
+
+int npp_implied_packing(NPP *npp, NPPROW *row, int which,
+      NPPCOL *var[], char set[])
+{     struct elem *ptr, *e, *i, *k;
+      int len = 0;
+      double b, eps;
+      /* build inequality (3) */
+      if (which == 0)
+      {  ptr = copy_form(npp, row, -1.0);
+         xassert(row->lb != -DBL_MAX);
+         b = - row->lb;
+      }
+      else if (which == 1)
+      {  ptr = copy_form(npp, row, +1.0);
+         xassert(row->ub != +DBL_MAX);
+         b = + row->ub;
+      }
+      /* remove non-binary variables to build relaxed inequality (5);
+         compute its right-hand side b~ with formula (6) */
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (!(e->xj->is_int && e->xj->lb == 0.0 && e->xj->ub == 1.0))
+         {  /* x[j] is non-binary variable */
+            if (e->aj > 0.0)
+            {  if (e->xj->lb == -DBL_MAX) goto done;
+               b -= e->aj * e->xj->lb;
+            }
+            else /* e->aj < 0.0 */
+            {  if (e->xj->ub == +DBL_MAX) goto done;
+               b -= e->aj * e->xj->ub;
+            }
+            /* a[j] = 0 means that variable x[j] is removed */
+            e->aj = 0.0;
+         }
+      }
+      /* substitute x[j] = 1 - x~[j] to build knapsack inequality (8);
+         compute its right-hand side beta with formula (11) */
+      for (e = ptr; e != NULL; e = e->next)
+         if (e->aj < 0.0) b -= e->aj;
+      /* if beta is close to zero, the knapsack inequality is either
+         infeasible or forcing inequality; this must never happen, so
+         we skip further analysis */
+      if (b < 1e-3) goto done;
+      /* build set P as well as sets Jp and Jn, and determine x[k] as
+         explained above in comments to the routine */
+      eps = 1e-3 + 1e-6 * b;
+      i = k = NULL;
+      for (e = ptr; e != NULL; e = e->next)
+      {  /* note that alfa[j] = |a[j]| */
+         if (fabs(e->aj) > 0.5 * (b + eps))
+         {  /* alfa[j] > (b + eps) / 2; include x[j] in set P, i.e. in
+               set Jp or Jn */
+            var[++len] = e->xj;
+            set[len] = (char)(e->aj > 0.0 ? 0 : 1);
+            /* alfa[i] = min alfa[j] over all j included in set P */
+            if (i == NULL || fabs(i->aj) > fabs(e->aj)) i = e;
+         }
+         else if (fabs(e->aj) >= 1e-3)
+         {  /* alfa[k] = max alfa[j] over all j not included in set P;
+               we skip coefficient a[j] if it is close to zero to avoid
+               numerically unreliable results */
+            if (k == NULL || fabs(k->aj) < fabs(e->aj)) k = e;
+         }
+      }
+      /* if alfa[k] satisfies to condition (13) for all j in P, include
+         x[k] in P */
+      if (i != NULL && k != NULL && fabs(i->aj) + fabs(k->aj) > b + eps)
+      {  var[++len] = k->xj;
+         set[len] = (char)(k->aj > 0.0 ? 0 : 1);
+      }
+      /* trivial packing inequality being redundant must never appear,
+         so we just ignore it */
+      if (len < 2) len = 0;
+done: drop_form(npp, ptr);
+      return len;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_is_covering - test if constraint is covering inequality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_is_covering(NPP *npp, NPPROW *row);
+*
+*  RETURNS
+*
+*  If the specified row (constraint) is covering inequality (see below),
+*  the routine npp_is_covering returns non-zero. Otherwise, it returns
+*  zero.
+*
+*  COVERING INEQUALITIES
+*
+*  In canonical format the covering inequality is the following:
+*
+*     sum  x[j] >= 1,                                                (1)
+*    j in J
+*
+*  where all variables x[j] are binary. This inequality expresses the
+*  condition that in any integer feasible solution variables in set J
+*  cannot be all equal to zero at the same time, i.e. at least one
+*  variable must take non-zero (unity) value. W.l.o.g. it is assumed
+*  that |J| >= 2, because if J is empty, the inequality (1) is
+*  infeasible, and if |J| = 1, the inequality (1) is a forcing row.
+*
+*  In general case the covering inequality may include original
+*  variables x[j] as well as their complements x~[j]:
+*
+*     sum   x[j] + sum   x~[j] >= 1,                                 (2)
+*    j in Jp      j in Jn
+*
+*  where Jp and Jn are not intersected. Therefore, using substitution
+*  x~[j] = 1 - x[j] gives the packing inequality in generalized format:
+*
+*     sum   x[j] - sum   x[j] >= 1 - |Jn|.                           (3)
+*    j in Jp      j in Jn
+*
+*  (May note that the inequality (3) cuts off infeasible solutions,
+*  where x[j] = 0 for all j in Jp and x[j] = 1 for all j in Jn.)
+*
+*  NOTE: If |J| = 2, the inequality (3) is equivalent to packing
+*        inequality (see the routine npp_is_packing). */
+
+int npp_is_covering(NPP *npp, NPPROW *row)
+{     /* test if constraint is covering inequality */
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int b;
+      xassert(npp == npp);
+      if (!(row->lb != -DBL_MAX && row->ub == +DBL_MAX))
+         return 0;
+      b = 1;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  col = aij->col;
+         if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+            return 0;
+         if (aij->val == +1.0)
+            ;
+         else if (aij->val == -1.0)
+            b--;
+         else
+            return 0;
+      }
+      if (row->lb != (double)b) return 0;
+      return 1;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_hidden_covering - identify hidden covering inequality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_hidden_covering(NPP *npp, NPPROW *row);
+*
+*  DESCRIPTION
+*
+*  The routine npp_hidden_covering processes specified inequality
+*  constraint, which includes only binary variables, and the number of
+*  the variables is not less than three. If the original inequality is
+*  equivalent to a covering inequality (see below), the routine
+*  replaces it by the equivalent inequality. If the original constraint
+*  is double-sided inequality, it is replaced by a pair of single-sided
+*  inequalities, if necessary.
+*
+*  RETURNS
+*
+*  If the original inequality constraint was replaced by equivalent
+*  covering inequality, the routine npp_hidden_covering returns
+*  non-zero. Otherwise, it returns zero.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  Consider an inequality constraint:
+*
+*     sum  a[j] x[j] >= b,                                           (1)
+*    j in J
+*
+*  where all variables x[j] are binary, and |J| >= 3. (In case of '<='
+*  inequality it can be transformed to '>=' format by multiplying both
+*  its sides by -1.)
+*
+*  Let Jp = {j: a[j] > 0}, Jn = {j: a[j] < 0}. Performing substitution
+*  x[j] = 1 - x~[j] for all j in Jn, we have:
+*
+*     sum   a[j] x[j] >= b  ==>
+*    j in J
+*
+*     sum   a[j] x[j] + sum   a[j] x[j] >= b  ==>
+*    j in Jp           j in Jn
+*
+*     sum   a[j] x[j] + sum   a[j] (1 - x~[j]) >= b  ==>
+*    j in Jp           j in Jn
+*
+*     sum  m   a[j] x[j] - sum   a[j] x~[j] >= b - sum   a[j].
+*    j in Jp              j in Jn                 j in Jn
+*
+*  Thus, meaning the transformation above, we can assume that in
+*  inequality (1) all coefficients a[j] are positive. Moreover, we can
+*  assume that b > 0, because otherwise the inequality (1) would be
+*  redundant (see the routine npp_analyze_row). It is then obvious that
+*  constraint (1) is equivalent to covering inequality only if:
+*
+*     a[j] >= b,                                                     (2)
+*
+*  for all j in J.
+*
+*  Once the original inequality (1) is replaced by equivalent covering
+*  inequality, we need to perform back substitution x~[j] = 1 - x[j] for
+*  all j in Jn (see above).
+*
+*  RECOVERING SOLUTION
+*
+*  None needed. */
+
+static int hidden_covering(NPP *npp, struct elem *ptr, double *_b)
+{     /* process inequality constraint: sum a[j] x[j] >= b;
+         0 - specified row is NOT hidden covering inequality;
+         1 - specified row is covering inequality;
+         2 - specified row is hidden covering inequality. */
+      struct elem *e;
+      int neg;
+      double b = *_b, eps;
+      xassert(npp == npp);
+      /* a[j] must be non-zero, x[j] must be binary, for all j in J */
+      for (e = ptr; e != NULL; e = e->next)
+      {  xassert(e->aj != 0.0);
+         xassert(e->xj->is_int);
+         xassert(e->xj->lb == 0.0 && e->xj->ub == 1.0);
+      }
+      /* check if the specified inequality constraint already has the
+         form of covering inequality */
+      neg = 0; /* neg is |Jn| */
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (e->aj == +1.0)
+            ;
+         else if (e->aj == -1.0)
+            neg++;
+         else
+            break;
+      }
+      if (e == NULL)
+      {  /* all coefficients a[j] are +1 or -1; check rhs b */
+         if (b == (double)(1 - neg))
+         {  /* it is covering inequality; no processing is needed */
+            return 1;
+         }
+      }
+      /* substitute x[j] = 1 - x~[j] for all j in Jn to make all a[j]
+         positive; the result is a~[j] = |a[j]| and new rhs b */
+      for (e = ptr; e != NULL; e = e->next)
+         if (e->aj < 0) b -= e->aj;
+      /* now a[j] > 0 for all j in J (actually |a[j]| are used) */
+      /* if b <= 0, skip processing--this case must not appear */
+      if (b < 1e-3) return 0;
+      /* now a[j] > 0 for all j in J, and b > 0 */
+      /* the specified constraint is equivalent to covering inequality
+         iff a[j] >= b for all j in J */
+      eps = 1e-9 + 1e-12 * fabs(b);
+      for (e = ptr; e != NULL; e = e->next)
+         if (fabs(e->aj) < b - eps) return 0;
+      /* perform back substitution x~[j] = 1 - x[j] and construct the
+         final equivalent covering inequality in generalized format */
+      b = 1.0;
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (e->aj > 0.0)
+            e->aj = +1.0;
+         else /* e->aj < 0.0 */
+            e->aj = -1.0, b -= 1.0;
+      }
+      *_b = b;
+      return 2;
+}
+
+int npp_hidden_covering(NPP *npp, NPPROW *row)
+{     /* identify hidden covering inequality */
+      NPPROW *copy;
+      NPPAIJ *aij;
+      struct elem *ptr, *e;
+      int kase, ret, count = 0;
+      double b;
+      /* the row must be inequality constraint */
+      xassert(row->lb < row->ub);
+      for (kase = 0; kase <= 1; kase++)
+      {  if (kase == 0)
+         {  /* process row lower bound */
+            if (row->lb == -DBL_MAX) continue;
+            ptr = copy_form(npp, row, +1.0);
+            b = + row->lb;
+         }
+         else
+         {  /* process row upper bound */
+            if (row->ub == +DBL_MAX) continue;
+            ptr = copy_form(npp, row, -1.0);
+            b = - row->ub;
+         }
+         /* now the inequality has the form "sum a[j] x[j] >= b" */
+         ret = hidden_covering(npp, ptr, &b);
+         xassert(0 <= ret && ret <= 2);
+         if (kase == 1 && ret == 1 || ret == 2)
+         {  /* the original inequality has been identified as hidden
+               covering inequality */
+            count++;
+#ifdef GLP_DEBUG
+            xprintf("Original constraint:\n");
+            for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+               xprintf(" %+g x%d", aij->val, aij->col->j);
+            if (row->lb != -DBL_MAX) xprintf(", >= %g", row->lb);
+            if (row->ub != +DBL_MAX) xprintf(", <= %g", row->ub);
+            xprintf("\n");
+            xprintf("Equivalent covering inequality:\n");
+            for (e = ptr; e != NULL; e = e->next)
+               xprintf(" %sx%d", e->aj > 0.0 ? "+" : "-", e->xj->j);
+            xprintf(", >= %g\n", b);
+#endif
+            if (row->lb == -DBL_MAX || row->ub == +DBL_MAX)
+            {  /* the original row is single-sided inequality; no copy
+                  is needed */
+               copy = NULL;
+            }
+            else
+            {  /* the original row is double-sided inequality; we need
+                  to create its copy for other bound before replacing it
+                  with the equivalent inequality */
+               copy = npp_add_row(npp);
+               if (kase == 0)
+               {  /* the copy is for upper bound */
+                  copy->lb = -DBL_MAX, copy->ub = row->ub;
+               }
+               else
+               {  /* the copy is for lower bound */
+                  copy->lb = row->lb, copy->ub = +DBL_MAX;
+               }
+               /* copy original row coefficients */
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  npp_add_aij(npp, copy, aij->col, aij->val);
+            }
+            /* replace the original inequality by equivalent one */
+            npp_erase_row(npp, row);
+            row->lb = b, row->ub = +DBL_MAX;
+            for (e = ptr; e != NULL; e = e->next)
+               npp_add_aij(npp, row, e->xj, e->aj);
+            /* continue processing upper bound for the copy */
+            if (copy != NULL) row = copy;
+         }
+         drop_form(npp, ptr);
+      }
+      return count;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_is_partitioning - test if constraint is partitioning equality
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_is_partitioning(NPP *npp, NPPROW *row);
+*
+*  RETURNS
+*
+*  If the specified row (constraint) is partitioning equality (see
+*  below), the routine npp_is_partitioning returns non-zero. Otherwise,
+*  it returns zero.
+*
+*  PARTITIONING EQUALITIES
+*
+*  In canonical format the partitioning equality is the following:
+*
+*     sum  x[j] = 1,                                                 (1)
+*    j in J
+*
+*  where all variables x[j] are binary. This equality expresses the
+*  condition that in any integer feasible solution exactly one variable
+*  in set J must take non-zero (unity) value while other variables must
+*  be equal to zero. W.l.o.g. it is assumed that |J| >= 2, because if
+*  J is empty, the inequality (1) is infeasible, and if |J| = 1, the
+*  inequality (1) is a fixing row.
+*
+*  In general case the partitioning equality may include original
+*  variables x[j] as well as their complements x~[j]:
+*
+*     sum   x[j] + sum   x~[j] = 1,                                  (2)
+*    j in Jp      j in Jn
+*
+*  where Jp and Jn are not intersected. Therefore, using substitution
+*  x~[j] = 1 - x[j] leads to the partitioning equality in generalized
+*  format:
+*
+*     sum   x[j] - sum   x[j] = 1 - |Jn|.                            (3)
+*    j in Jp      j in Jn */
+
+int npp_is_partitioning(NPP *npp, NPPROW *row)
+{     /* test if constraint is partitioning equality */
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int b;
+      xassert(npp == npp);
+      if (row->lb != row->ub) return 0;
+      b = 1;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  col = aij->col;
+         if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+            return 0;
+         if (aij->val == +1.0)
+            ;
+         else if (aij->val == -1.0)
+            b--;
+         else
+            return 0;
+      }
+      if (row->lb != (double)b) return 0;
+      return 1;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_reduce_ineq_coef - reduce inequality constraint coefficients
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_reduce_ineq_coef(NPP *npp, NPPROW *row);
+*
+*  DESCRIPTION
+*
+*  The routine npp_reduce_ineq_coef processes specified inequality
+*  constraint attempting to replace it by an equivalent constraint,
+*  where magnitude of coefficients at binary variables is smaller than
+*  in the original constraint. If the inequality is double-sided, it is
+*  replaced by a pair of single-sided inequalities, if necessary.
+*
+*  RETURNS
+*
+*  The routine npp_reduce_ineq_coef returns the number of coefficients
+*  reduced.
+*
+*  BACKGROUND
+*
+*  Consider an inequality constraint:
+*
+*     sum  a[j] x[j] >= b.                                           (1)
+*    j in J
+*
+*  (In case of '<=' inequality it can be transformed to '>=' format by
+*  multiplying both its sides by -1.) Let x[k] be a binary variable;
+*  other variables can be integer as well as continuous. We can write
+*  constraint (1) as follows:
+*
+*     a[k] x[k] + t[k] >= b,                                         (2)
+*
+*  where:
+*
+*     t[k] = sum      a[j] x[j].                                     (3)
+*           j in J\{k}
+*
+*  Since x[k] is binary, constraint (2) is equivalent to disjunction of
+*  the following two constraints:
+*
+*     x[k] = 0,  t[k] >= b                                           (4)
+*
+*        OR
+*
+*     x[k] = 1,  t[k] >= b - a[k].                                   (5)
+*
+*  Let also that for the partial sum t[k] be known some its implied
+*  lower bound inf t[k].
+*
+*  Case a[k] > 0. Let inf t[k] < b, since otherwise both constraints
+*  (4) and (5) and therefore constraint (2) are redundant.
+*  If inf t[k] > b - a[k], only constraint (5) is redundant, in which
+*  case it can be replaced with the following redundant and therefore
+*  equivalent constraint:
+*
+*     t[k] >= b - a'[k] = inf t[k],                                  (6)
+*
+*  where:
+*
+*     a'[k] = b - inf t[k].                                          (7)
+*
+*  Thus, the original constraint (2) is equivalent to the following
+*  constraint with coefficient at variable x[k] changed:
+*
+*     a'[k] x[k] + t[k] >= b.                                        (8)
+*
+*  From inf t[k] < b it follows that a'[k] > 0, i.e. the coefficient
+*  at x[k] keeps its sign. And from inf t[k] > b - a[k] it follows that
+*  a'[k] < a[k], i.e. the coefficient reduces in magnitude.
+*
+*  Case a[k] < 0. Let inf t[k] < b - a[k], since otherwise both
+*  constraints (4) and (5) and therefore constraint (2) are redundant.
+*  If inf t[k] > b, only constraint (4) is redundant, in which case it
+*  can be replaced with the following redundant and therefore equivalent
+*  constraint:
+*
+*     t[k] >= b' = inf t[k].                                         (9)
+*
+*  Rewriting constraint (5) as follows:
+*
+*     t[k] >= b - a[k] = b' - a'[k],                                (10)
+*
+*  where:
+*
+*     a'[k] = a[k] + b' - b = a[k] + inf t[k] - b,                  (11)
+*
+*  we can see that disjunction of constraint (9) and (10) is equivalent
+*  to disjunction of constraint (4) and (5), from which it follows that
+*  the original constraint (2) is equivalent to the following constraint
+*  with both coefficient at variable x[k] and right-hand side changed:
+*
+*     a'[k] x[k] + t[k] >= b'.                                      (12)
+*
+*  From inf t[k] < b - a[k] it follows that a'[k] < 0, i.e. the
+*  coefficient at x[k] keeps its sign. And from inf t[k] > b it follows
+*  that a'[k] > a[k], i.e. the coefficient reduces in magnitude.
+*
+*  PROBLEM TRANSFORMATION
+*
+*  In the routine npp_reduce_ineq_coef the following implied lower
+*  bound of the partial sum (3) is used:
+*
+*     inf t[k] = sum       a[j] l[j] + sum       a[j] u[j],         (13)
+*               j in Jp\{k}           k in Jn\{k}
+*
+*  where Jp = {j : a[j] > 0}, Jn = {j : a[j] < 0}, l[j] and u[j] are
+*  lower and upper bounds, resp., of variable x[j].
+*
+*  In order to compute inf t[k] more efficiently, the following formula,
+*  which is equivalent to (13), is actually used:
+*
+*                ( h - a[k] l[k] = h,        if a[k] > 0,
+*     inf t[k] = <                                                  (14)
+*                ( h - a[k] u[k] = h - a[k], if a[k] < 0,
+*
+*  where:
+*
+*     h = sum   a[j] l[j] + sum   a[j] u[j]                         (15)
+*        j in Jp           j in Jn
+*
+*  is the implied lower bound of row (1).
+*
+*  Reduction of positive coefficient (a[k] > 0) does not change value
+*  of h, since l[k] = 0. In case of reduction of negative coefficient
+*  (a[k] < 0) from (11) it follows that:
+*
+*     delta a[k] = a'[k] - a[k] = inf t[k] - b  (> 0),              (16)
+*
+*  so new value of h (accounting that u[k] = 1) can be computed as
+*  follows:
+*
+*     h := h + delta a[k] = h + (inf t[k] - b).                     (17)
+*
+*  RECOVERING SOLUTION
+*
+*  None needed. */
+
+static int reduce_ineq_coef(NPP *npp, struct elem *ptr, double *_b)
+{     /* process inequality constraint: sum a[j] x[j] >= b */
+      /* returns: the number of coefficients reduced */
+      struct elem *e;
+      int count = 0;
+      double h, inf_t, new_a, b = *_b;
+      xassert(npp == npp);
+      /* compute h; see (15) */
+      h = 0.0;
+      for (e = ptr; e != NULL; e = e->next)
+      {  if (e->aj > 0.0)
+         {  if (e->xj->lb == -DBL_MAX) goto done;
+            h += e->aj * e->xj->lb;
+         }
+         else /* e->aj < 0.0 */
+         {  if (e->xj->ub == +DBL_MAX) goto done;
+            h += e->aj * e->xj->ub;
+         }
+      }
+      /* perform reduction of coefficients at binary variables */
+      for (e = ptr; e != NULL; e = e->next)
+      {  /* skip non-binary variable */
+         if (!(e->xj->is_int && e->xj->lb == 0.0 && e->xj->ub == 1.0))
+            continue;
+         if (e->aj > 0.0)
+         {  /* compute inf t[k]; see (14) */
+            inf_t = h;
+            if (b - e->aj < inf_t && inf_t < b)
+            {  /* compute reduced coefficient a'[k]; see (7) */
+               new_a = b - inf_t;
+               if (new_a >= +1e-3 &&
+                   e->aj - new_a >= 0.01 * (1.0 + e->aj))
+               {  /* accept a'[k] */
+#ifdef GLP_DEBUG
+                  xprintf("+");
+#endif
+                  e->aj = new_a;
+                  count++;
+               }
+            }
+         }
+         else /* e->aj < 0.0 */
+         {  /* compute inf t[k]; see (14) */
+            inf_t = h - e->aj;
+            if (b < inf_t && inf_t < b - e->aj)
+            {  /* compute reduced coefficient a'[k]; see (11) */
+               new_a = e->aj + (inf_t - b);
+               if (new_a <= -1e-3 &&
+                   new_a - e->aj >= 0.01 * (1.0 - e->aj))
+               {  /* accept a'[k] */
+#ifdef GLP_DEBUG
+                  xprintf("-");
+#endif
+                  e->aj = new_a;
+                  /* update h; see (17) */
+                  h += (inf_t - b);
+                  /* compute b'; see (9) */
+                  b = inf_t;
+                  count++;
+               }
+            }
+         }
+      }
+      *_b = b;
+done: return count;
+}
+
+int npp_reduce_ineq_coef(NPP *npp, NPPROW *row)
+{     /* reduce inequality constraint coefficients */
+      NPPROW *copy;
+      NPPAIJ *aij;
+      struct elem *ptr, *e;
+      int kase, count[2];
+      double b;
+      /* the row must be inequality constraint */
+      xassert(row->lb < row->ub);
+      count[0] = count[1] = 0;
+      for (kase = 0; kase <= 1; kase++)
+      {  if (kase == 0)
+         {  /* process row lower bound */
+            if (row->lb == -DBL_MAX) continue;
+#ifdef GLP_DEBUG
+            xprintf("L");
+#endif
+            ptr = copy_form(npp, row, +1.0);
+            b = + row->lb;
+         }
+         else
+         {  /* process row upper bound */
+            if (row->ub == +DBL_MAX) continue;
+#ifdef GLP_DEBUG
+            xprintf("U");
+#endif
+            ptr = copy_form(npp, row, -1.0);
+            b = - row->ub;
+         }
+         /* now the inequality has the form "sum a[j] x[j] >= b" */
+         count[kase] = reduce_ineq_coef(npp, ptr, &b);
+         if (count[kase] > 0)
+         {  /* the original inequality has been replaced by equivalent
+               one with coefficients reduced */
+            if (row->lb == -DBL_MAX || row->ub == +DBL_MAX)
+            {  /* the original row is single-sided inequality; no copy
+                  is needed */
+               copy = NULL;
+            }
+            else
+            {  /* the original row is double-sided inequality; we need
+                  to create its copy for other bound before replacing it
+                  with the equivalent inequality */
+#ifdef GLP_DEBUG
+               xprintf("*");
+#endif
+               copy = npp_add_row(npp);
+               if (kase == 0)
+               {  /* the copy is for upper bound */
+                  copy->lb = -DBL_MAX, copy->ub = row->ub;
+               }
+               else
+               {  /* the copy is for lower bound */
+                  copy->lb = row->lb, copy->ub = +DBL_MAX;
+               }
+               /* copy original row coefficients */
+               for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                  npp_add_aij(npp, copy, aij->col, aij->val);
+            }
+            /* replace the original inequality by equivalent one */
+            npp_erase_row(npp, row);
+            row->lb = b, row->ub = +DBL_MAX;
+            for (e = ptr; e != NULL; e = e->next)
+               npp_add_aij(npp, row, e->xj, e->aj);
+            /* continue processing upper bound for the copy */
+            if (copy != NULL) row = copy;
+         }
+         drop_form(npp, ptr);
+      }
+      return count[0] + count[1];
+}
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp5.c b/test/monniaux/glpk-4.65/src/npp/npp5.c
new file mode 100644
index 00000000..2fad496d
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp5.c
@@ -0,0 +1,809 @@
+/* npp5.c */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2009-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+/***********************************************************************
+*  NAME
+*
+*  npp_clean_prob - perform initial LP/MIP processing
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  void npp_clean_prob(NPP *npp);
+*
+*  DESCRIPTION
+*
+*  The routine npp_clean_prob performs initial LP/MIP processing that
+*  currently includes:
+*
+*  1) removing free rows;
+*
+*  2) replacing double-sided constraint rows with almost identical
+*     bounds, by equality constraint rows;
+*
+*  3) removing fixed columns;
+*
+*  4) replacing double-bounded columns with almost identical bounds by
+*     fixed columns and removing those columns;
+*
+*  5) initial processing constraint coefficients (not implemented);
+*
+*  6) initial processing objective coefficients (not implemented). */
+
+void npp_clean_prob(NPP *npp)
+{     /* perform initial LP/MIP processing */
+      NPPROW *row, *next_row;
+      NPPCOL *col, *next_col;
+      int ret;
+      xassert(npp == npp);
+      /* process rows which originally are free */
+      for (row = npp->r_head; row != NULL; row = next_row)
+      {  next_row = row->next;
+         if (row->lb == -DBL_MAX && row->ub == +DBL_MAX)
+         {  /* process free row */
+#ifdef GLP_DEBUG
+            xprintf("1");
+#endif
+            npp_free_row(npp, row);
+            /* row was deleted */
+         }
+      }
+      /* process rows which originally are double-sided inequalities */
+      for (row = npp->r_head; row != NULL; row = next_row)
+      {  next_row = row->next;
+         if (row->lb != -DBL_MAX && row->ub != +DBL_MAX &&
+             row->lb < row->ub)
+         {  ret = npp_make_equality(npp, row);
+            if (ret == 0)
+               ;
+            else if (ret == 1)
+            {  /* row was replaced by equality constraint */
+#ifdef GLP_DEBUG
+               xprintf("2");
+#endif
+            }
+            else
+               xassert(ret != ret);
+         }
+      }
+      /* process columns which are originally fixed */
+      for (col = npp->c_head; col != NULL; col = next_col)
+      {  next_col = col->next;
+         if (col->lb == col->ub)
+         {  /* process fixed column */
+#ifdef GLP_DEBUG
+            xprintf("3");
+#endif
+            npp_fixed_col(npp, col);
+            /* column was deleted */
+         }
+      }
+      /* process columns which are originally double-bounded */
+      for (col = npp->c_head; col != NULL; col = next_col)
+      {  next_col = col->next;
+         if (col->lb != -DBL_MAX && col->ub != +DBL_MAX &&
+             col->lb < col->ub)
+         {  ret = npp_make_fixed(npp, col);
+            if (ret == 0)
+               ;
+            else if (ret == 1)
+            {  /* column was replaced by fixed column; process it */
+#ifdef GLP_DEBUG
+               xprintf("4");
+#endif
+               npp_fixed_col(npp, col);
+               /* column was deleted */
+            }
+         }
+      }
+      return;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_process_row - perform basic row processing
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_process_row(NPP *npp, NPPROW *row, int hard);
+*
+*  DESCRIPTION
+*
+*  The routine npp_process_row performs basic row processing that
+*  currently includes:
+*
+*  1) removing empty row;
+*
+*  2) removing equality constraint row singleton and corresponding
+*     column;
+*
+*  3) removing inequality constraint row singleton and corresponding
+*     column if it was fixed;
+*
+*  4) performing general row analysis;
+*
+*  5) removing redundant row bounds;
+*
+*  6) removing forcing row and corresponding columns;
+*
+*  7) removing row which becomes free due to redundant bounds;
+*
+*  8) computing implied bounds for all columns in the row and using
+*     them to strengthen current column bounds (MIP only, optional,
+*     performed if the flag hard is on).
+*
+*  Additionally the routine may activate affected rows and/or columns
+*  for further processing.
+*
+*  RETURNS
+*
+*  0           success;
+*
+*  GLP_ENOPFS  primal/integer infeasibility detected;
+*
+*  GLP_ENODFS  dual infeasibility detected. */
+
+int npp_process_row(NPP *npp, NPPROW *row, int hard)
+{     /* perform basic row processing */
+      NPPCOL *col;
+      NPPAIJ *aij, *next_aij, *aaa;
+      int ret;
+      /* row must not be free */
+      xassert(!(row->lb == -DBL_MAX && row->ub == +DBL_MAX));
+      /* start processing row */
+      if (row->ptr == NULL)
+      {  /* empty row */
+         ret = npp_empty_row(npp, row);
+         if (ret == 0)
+         {  /* row was deleted */
+#ifdef GLP_DEBUG
+            xprintf("A");
+#endif
+            return 0;
+         }
+         else if (ret == 1)
+         {  /* primal infeasibility */
+            return GLP_ENOPFS;
+         }
+         else
+            xassert(ret != ret);
+      }
+      if (row->ptr->r_next == NULL)
+      {  /* row singleton */
+         col = row->ptr->col;
+         if (row->lb == row->ub)
+         {  /* equality constraint */
+            ret = npp_eq_singlet(npp, row);
+            if (ret == 0)
+            {  /* column was fixed, row was deleted */
+#ifdef GLP_DEBUG
+               xprintf("B");
+#endif
+               /* activate rows affected by column */
+               for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+                  npp_activate_row(npp, aij->row);
+               /* process fixed column */
+               npp_fixed_col(npp, col);
+               /* column was deleted */
+               return 0;
+            }
+            else if (ret == 1 || ret == 2)
+            {  /* primal/integer infeasibility */
+               return GLP_ENOPFS;
+            }
+            else
+               xassert(ret != ret);
+         }
+         else
+         {  /* inequality constraint */
+            ret = npp_ineq_singlet(npp, row);
+            if (0 <= ret && ret <= 3)
+            {  /* row was deleted */
+#ifdef GLP_DEBUG
+               xprintf("C");
+#endif
+               /* activate column, since its length was changed due to
+                  row deletion */
+               npp_activate_col(npp, col);
+               if (ret >= 2)
+               {  /* column bounds changed significantly or column was
+                     fixed */
+                  /* activate rows affected by column */
+                  for (aij = col->ptr; aij != NULL; aij = aij->c_next)
+                     npp_activate_row(npp, aij->row);
+               }
+               if (ret == 3)
+               {  /* column was fixed; process it */
+#ifdef GLP_DEBUG
+                  xprintf("D");
+#endif
+                  npp_fixed_col(npp, col);
+                  /* column was deleted */
+               }
+               return 0;
+            }
+            else if (ret == 4)
+            {  /* primal infeasibility */
+               return GLP_ENOPFS;
+            }
+            else
+               xassert(ret != ret);
+         }
+      }
+#if 0
+      /* sometimes this causes too large round-off errors; probably
+         pivot coefficient should be chosen more carefully */
+      if (row->ptr->r_next->r_next == NULL)
+      {  /* row doubleton */
+         if (row->lb == row->ub)
+         {  /* equality constraint */
+            if (!(row->ptr->col->is_int ||
+                  row->ptr->r_next->col->is_int))
+            {  /* both columns are continuous */
+               NPPCOL *q;
+               q = npp_eq_doublet(npp, row);
+               if (q != NULL)
+               {  /* column q was eliminated */
+#ifdef GLP_DEBUG
+                  xprintf("E");
+#endif
+                  /* now column q is singleton of type "implied slack
+                     variable"; we process it here to make sure that on
+                     recovering basic solution the row is always active
+                     equality constraint (as required by the routine
+                     rcv_eq_doublet) */
+                  xassert(npp_process_col(npp, q) == 0);
+                  /* column q was deleted; note that row p also may be
+                     deleted */
+                  return 0;
+               }
+            }
+         }
+      }
+#endif
+      /* general row analysis */
+      ret = npp_analyze_row(npp, row);
+      xassert(0x00 <= ret && ret <= 0xFF);
+      if (ret == 0x33)
+      {  /* row bounds are inconsistent with column bounds */
+         return GLP_ENOPFS;
+      }
+      if ((ret & 0x0F) == 0x00)
+      {  /* row lower bound does not exist or redundant */
+         if (row->lb != -DBL_MAX)
+         {  /* remove redundant row lower bound */
+#ifdef GLP_DEBUG
+            xprintf("F");
+#endif
+            npp_inactive_bound(npp, row, 0);
+         }
+      }
+      else if ((ret & 0x0F) == 0x01)
+      {  /* row lower bound can be active */
+         /* see below */
+      }
+      else if ((ret & 0x0F) == 0x02)
+      {  /* row lower bound is a forcing bound */
+#ifdef GLP_DEBUG
+         xprintf("G");
+#endif
+         /* process forcing row */
+         if (npp_forcing_row(npp, row, 0) == 0)
+fixup:   {  /* columns were fixed, row was made free */
+            for (aij = row->ptr; aij != NULL; aij = next_aij)
+            {  /* process column fixed by forcing row */
+#ifdef GLP_DEBUG
+               xprintf("H");
+#endif
+               col = aij->col;
+               next_aij = aij->r_next;
+               /* activate rows affected by column */
+               for (aaa = col->ptr; aaa != NULL; aaa = aaa->c_next)
+                  npp_activate_row(npp, aaa->row);
+               /* process fixed column */
+               npp_fixed_col(npp, col);
+               /* column was deleted */
+            }
+            /* process free row (which now is empty due to deletion of
+               all its columns) */
+            npp_free_row(npp, row);
+            /* row was deleted */
+            return 0;
+         }
+      }
+      else
+         xassert(ret != ret);
+      if ((ret & 0xF0) == 0x00)
+      {  /* row upper bound does not exist or redundant */
+         if (row->ub != +DBL_MAX)
+         {  /* remove redundant row upper bound */
+#ifdef GLP_DEBUG
+            xprintf("I");
+#endif
+            npp_inactive_bound(npp, row, 1);
+         }
+      }
+      else if ((ret & 0xF0) == 0x10)
+      {  /* row upper bound can be active */
+         /* see below */
+      }
+      else if ((ret & 0xF0) == 0x20)
+      {  /* row upper bound is a forcing bound */
+#ifdef GLP_DEBUG
+         xprintf("J");
+#endif
+         /* process forcing row */
+         if (npp_forcing_row(npp, row, 1) == 0) goto fixup;
+      }
+      else
+         xassert(ret != ret);
+      if (row->lb == -DBL_MAX && row->ub == +DBL_MAX)
+      {  /* row became free due to redundant bounds removal */
+#ifdef GLP_DEBUG
+         xprintf("K");
+#endif
+         /* activate its columns, since their length will change due
+            to row deletion */
+         for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+            npp_activate_col(npp, aij->col);
+         /* process free row */
+         npp_free_row(npp, row);
+         /* row was deleted */
+         return 0;
+      }
+#if 1 /* 23/XII-2009 */
+      /* row lower and/or upper bounds can be active */
+      if (npp->sol == GLP_MIP && hard)
+      {  /* improve current column bounds (optional) */
+         if (npp_improve_bounds(npp, row, 1) < 0)
+            return GLP_ENOPFS;
+      }
+#endif
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_improve_bounds - improve current column bounds
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_improve_bounds(NPP *npp, NPPROW *row, int flag);
+*
+*  DESCRIPTION
+*
+*  The routine npp_improve_bounds analyzes specified row (inequality
+*  or equality constraint) to determine implied column bounds and then
+*  uses these bounds to improve (strengthen) current column bounds.
+*
+*  If the flag is on and current column bounds changed significantly
+*  or the column was fixed, the routine activate rows affected by the
+*  column for further processing. (This feature is intended to be used
+*  in the main loop of the routine npp_process_row.)
+*
+*  NOTE: This operation can be used for MIP problem only.
+*
+*  RETURNS
+*
+*  The routine npp_improve_bounds returns the number of significantly
+*  changed bounds plus the number of column having been fixed due to
+*  bound improvements. However, if the routine detects primal/integer
+*  infeasibility, it returns a negative value. */
+
+int npp_improve_bounds(NPP *npp, NPPROW *row, int flag)
+{     /* improve current column bounds */
+      NPPCOL *col;
+      NPPAIJ *aij, *next_aij, *aaa;
+      int kase, ret, count = 0;
+      double lb, ub;
+      xassert(npp->sol == GLP_MIP);
+      /* row must not be free */
+      xassert(!(row->lb == -DBL_MAX && row->ub == +DBL_MAX));
+      /* determine implied column bounds */
+      npp_implied_bounds(npp, row);
+      /* and use these bounds to strengthen current column bounds */
+      for (aij = row->ptr; aij != NULL; aij = next_aij)
+      {  col = aij->col;
+         next_aij = aij->r_next;
+         for (kase = 0; kase <= 1; kase++)
+         {  /* save current column bounds */
+            lb = col->lb, ub = col->ub;
+            if (kase == 0)
+            {  /* process implied column lower bound */
+               if (col->ll.ll == -DBL_MAX) continue;
+               ret = npp_implied_lower(npp, col, col->ll.ll);
+            }
+            else
+            {  /* process implied column upper bound */
+               if (col->uu.uu == +DBL_MAX) continue;
+               ret = npp_implied_upper(npp, col, col->uu.uu);
+            }
+            if (ret == 0 || ret == 1)
+            {  /* current column bounds did not change or changed, but
+                  not significantly; restore current column bounds */
+               col->lb = lb, col->ub = ub;
+            }
+            else if (ret == 2 || ret == 3)
+            {  /* current column bounds changed significantly or column
+                  was fixed */
+#ifdef GLP_DEBUG
+               xprintf("L");
+#endif
+               count++;
+               /* activate other rows affected by column, if required */
+               if (flag)
+               {  for (aaa = col->ptr; aaa != NULL; aaa = aaa->c_next)
+                  {  if (aaa->row != row)
+                        npp_activate_row(npp, aaa->row);
+                  }
+               }
+               if (ret == 3)
+               {  /* process fixed column */
+#ifdef GLP_DEBUG
+                  xprintf("M");
+#endif
+                  npp_fixed_col(npp, col);
+                  /* column was deleted */
+                  break; /* for kase */
+               }
+            }
+            else if (ret == 4)
+            {  /* primal/integer infeasibility */
+               return -1;
+            }
+            else
+               xassert(ret != ret);
+         }
+      }
+      return count;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_process_col - perform basic column processing
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_process_col(NPP *npp, NPPCOL *col);
+*
+*  DESCRIPTION
+*
+*  The routine npp_process_col performs basic column processing that
+*  currently includes:
+*
+*  1) fixing and removing empty column;
+*
+*  2) removing column singleton, which is implied slack variable, and
+*     corresponding row if it becomes free;
+*
+*  3) removing bounds of column, which is implied free variable, and
+*     replacing corresponding row by equality constraint.
+*
+*  Additionally the routine may activate affected rows and/or columns
+*  for further processing.
+*
+*  RETURNS
+*
+*  0           success;
+*
+*  GLP_ENOPFS  primal/integer infeasibility detected;
+*
+*  GLP_ENODFS  dual infeasibility detected. */
+
+int npp_process_col(NPP *npp, NPPCOL *col)
+{     /* perform basic column processing */
+      NPPROW *row;
+      NPPAIJ *aij;
+      int ret;
+      /* column must not be fixed */
+      xassert(col->lb < col->ub);
+      /* start processing column */
+      if (col->ptr == NULL)
+      {  /* empty column */
+         ret = npp_empty_col(npp, col);
+         if (ret == 0)
+         {  /* column was fixed and deleted */
+#ifdef GLP_DEBUG
+            xprintf("N");
+#endif
+            return 0;
+         }
+         else if (ret == 1)
+         {  /* dual infeasibility */
+            return GLP_ENODFS;
+         }
+         else
+            xassert(ret != ret);
+      }
+      if (col->ptr->c_next == NULL)
+      {  /* column singleton */
+         row = col->ptr->row;
+         if (row->lb == row->ub)
+         {  /* equality constraint */
+            if (!col->is_int)
+slack:      {  /* implied slack variable */
+#ifdef GLP_DEBUG
+               xprintf("O");
+#endif
+               npp_implied_slack(npp, col);
+               /* column was deleted */
+               if (row->lb == -DBL_MAX && row->ub == +DBL_MAX)
+               {  /* row became free due to implied slack variable */
+#ifdef GLP_DEBUG
+                  xprintf("P");
+#endif
+                  /* activate columns affected by row */
+                  for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+                     npp_activate_col(npp, aij->col);
+                  /* process free row */
+                  npp_free_row(npp, row);
+                  /* row was deleted */
+               }
+               else
+               {  /* row became inequality constraint; activate it
+                     since its length changed due to column deletion */
+                  npp_activate_row(npp, row);
+               }
+               return 0;
+            }
+         }
+         else
+         {  /* inequality constraint */
+            if (!col->is_int)
+            {  ret = npp_implied_free(npp, col);
+               if (ret == 0)
+               {  /* implied free variable */
+#ifdef GLP_DEBUG
+                  xprintf("Q");
+#endif
+                  /* column bounds were removed, row was replaced by
+                     equality constraint */
+                  goto slack;
+               }
+               else if (ret == 1)
+               {  /* column is not implied free variable, because its
+                     lower and/or upper bounds can be active */
+               }
+               else if (ret == 2)
+               {  /* dual infeasibility */
+                  return GLP_ENODFS;
+               }
+            }
+         }
+      }
+      /* column still exists */
+      return 0;
+}
+
+/***********************************************************************
+*  NAME
+*
+*  npp_process_prob - perform basic LP/MIP processing
+*
+*  SYNOPSIS
+*
+*  #include "glpnpp.h"
+*  int npp_process_prob(NPP *npp, int hard);
+*
+*  DESCRIPTION
+*
+*  The routine npp_process_prob performs basic LP/MIP processing that
+*  currently includes:
+*
+*  1) initial LP/MIP processing (see the routine npp_clean_prob),
+*
+*  2) basic row processing (see the routine npp_process_row), and
+*
+*  3) basic column processing (see the routine npp_process_col).
+*
+*  If the flag hard is on, the routine attempts to improve current
+*  column bounds multiple times within the main processing loop, in
+*  which case this feature may take a time. Otherwise, if the flag hard
+*  is off, improving column bounds is performed only once at the end of
+*  the main loop. (Note that this feature is used for MIP only.)
+*
+*  The routine uses two sets: the set of active rows and the set of
+*  active columns. Rows/columns are marked by a flag (the field temp in
+*  NPPROW/NPPCOL). If the flag is non-zero, the row/column is active,
+*  in which case it is placed in the beginning of the row/column list;
+*  otherwise, if the flag is zero, the row/column is inactive, in which
+*  case it is placed in the end of the row/column list. If a row/column
+*  being currently processed may affect other rows/columns, the latters
+*  are activated for further processing.
+*
+*  RETURNS
+*
+*  0           success;
+*
+*  GLP_ENOPFS  primal/integer infeasibility detected;
+*
+*  GLP_ENODFS  dual infeasibility detected. */
+
+int npp_process_prob(NPP *npp, int hard)
+{     /* perform basic LP/MIP processing */
+      NPPROW *row;
+      NPPCOL *col;
+      int processing, ret;
+      /* perform initial LP/MIP processing */
+      npp_clean_prob(npp);
+      /* activate all remaining rows and columns */
+      for (row = npp->r_head; row != NULL; row = row->next)
+         row->temp = 1;
+      for (col = npp->c_head; col != NULL; col = col->next)
+         col->temp = 1;
+      /* main processing loop */
+      processing = 1;
+      while (processing)
+      {  processing = 0;
+         /* process all active rows */
+         for (;;)
+         {  row = npp->r_head;
+            if (row == NULL || !row->temp) break;
+            npp_deactivate_row(npp, row);
+            ret = npp_process_row(npp, row, hard);
+            if (ret != 0) goto done;
+            processing = 1;
+         }
+         /* process all active columns */
+         for (;;)
+         {  col = npp->c_head;
+            if (col == NULL || !col->temp) break;
+            npp_deactivate_col(npp, col);
+            ret = npp_process_col(npp, col);
+            if (ret != 0) goto done;
+            processing = 1;
+         }
+      }
+#if 1 /* 23/XII-2009 */
+      if (npp->sol == GLP_MIP && !hard)
+      {  /* improve current column bounds (optional) */
+         for (row = npp->r_head; row != NULL; row = row->next)
+         {  if (npp_improve_bounds(npp, row, 0) < 0)
+            {  ret = GLP_ENOPFS;
+               goto done;
+            }
+         }
+      }
+#endif
+      /* all seems ok */
+      ret = 0;
+done: xassert(ret == 0 || ret == GLP_ENOPFS || ret == GLP_ENODFS);
+#ifdef GLP_DEBUG
+      xprintf("\n");
+#endif
+      return ret;
+}
+
+/**********************************************************************/
+
+int npp_simplex(NPP *npp, const glp_smcp *parm)
+{     /* process LP prior to applying primal/dual simplex method */
+      int ret;
+      xassert(npp->sol == GLP_SOL);
+      xassert(parm == parm);
+      ret = npp_process_prob(npp, 0);
+      return ret;
+}
+
+/**********************************************************************/
+
+int npp_integer(NPP *npp, const glp_iocp *parm)
+{     /* process MIP prior to applying branch-and-bound method */
+      NPPROW *row, *prev_row;
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int count, ret;
+      xassert(npp->sol == GLP_MIP);
+      xassert(parm == parm);
+      /*==============================================================*/
+      /* perform basic MIP processing */
+      ret = npp_process_prob(npp, 1);
+      if (ret != 0) goto done;
+      /*==============================================================*/
+      /* binarize problem, if required */
+      if (parm->binarize)
+         npp_binarize_prob(npp);
+      /*==============================================================*/
+      /* identify hidden packing inequalities */
+      count = 0;
+      /* new rows will be added to the end of the row list, so we go
+         from the end to beginning of the row list */
+      for (row = npp->r_tail; row != NULL; row = prev_row)
+      {  prev_row = row->prev;
+         /* skip free row */
+         if (row->lb == -DBL_MAX && row->ub == +DBL_MAX) continue;
+         /* skip equality constraint */
+         if (row->lb == row->ub) continue;
+         /* skip row having less than two variables */
+         if (row->ptr == NULL || row->ptr->r_next == NULL) continue;
+         /* skip row having non-binary variables */
+         for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+         {  col = aij->col;
+            if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+               break;
+         }
+         if (aij != NULL) continue;
+         count += npp_hidden_packing(npp, row);
+      }
+      if (count > 0)
+         xprintf("%d hidden packing inequaliti(es) were detected\n",
+            count);
+      /*==============================================================*/
+      /* identify hidden covering inequalities */
+      count = 0;
+      /* new rows will be added to the end of the row list, so we go
+         from the end to beginning of the row list */
+      for (row = npp->r_tail; row != NULL; row = prev_row)
+      {  prev_row = row->prev;
+         /* skip free row */
+         if (row->lb == -DBL_MAX && row->ub == +DBL_MAX) continue;
+         /* skip equality constraint */
+         if (row->lb == row->ub) continue;
+         /* skip row having less than three variables */
+         if (row->ptr == NULL || row->ptr->r_next == NULL ||
+             row->ptr->r_next->r_next == NULL) continue;
+         /* skip row having non-binary variables */
+         for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+         {  col = aij->col;
+            if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+               break;
+         }
+         if (aij != NULL) continue;
+         count += npp_hidden_covering(npp, row);
+      }
+      if (count > 0)
+         xprintf("%d hidden covering inequaliti(es) were detected\n",
+            count);
+      /*==============================================================*/
+      /* reduce inequality constraint coefficients */
+      count = 0;
+      /* new rows will be added to the end of the row list, so we go
+         from the end to beginning of the row list */
+      for (row = npp->r_tail; row != NULL; row = prev_row)
+      {  prev_row = row->prev;
+         /* skip equality constraint */
+         if (row->lb == row->ub) continue;
+         count += npp_reduce_ineq_coef(npp, row);
+      }
+      if (count > 0)
+         xprintf("%d constraint coefficient(s) were reduced\n", count);
+      /*==============================================================*/
+#ifdef GLP_DEBUG
+      routine(npp);
+#endif
+      /*==============================================================*/
+      /* all seems ok */
+      ret = 0;
+done: return ret;
+}
+
+/* eof */
diff --git a/test/monniaux/glpk-4.65/src/npp/npp6.c b/test/monniaux/glpk-4.65/src/npp/npp6.c
new file mode 100644
index 00000000..b57f8615
--- /dev/null
+++ b/test/monniaux/glpk-4.65/src/npp/npp6.c
@@ -0,0 +1,1500 @@
+/* npp6.c (translate feasibility problem to CNF-SAT) */
+
+/***********************************************************************
+*  This code is part of GLPK (GNU Linear Programming Kit).
+*
+*  Copyright (C) 2011-2017 Andrew Makhorin, Department for Applied
+*  Informatics, Moscow Aviation Institute, Moscow, Russia. All rights
+*  reserved. E-mail: <mao@gnu.org>.
+*
+*  GLPK is free software: you can redistribute it and/or modify it
+*  under the terms of the GNU General Public License as published by
+*  the Free Software Foundation, either version 3 of the License, or
+*  (at your option) any later version.
+*
+*  GLPK is distributed in the hope that it will be useful, but WITHOUT
+*  ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
+*  or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public
+*  License for more details.
+*
+*  You should have received a copy of the GNU General Public License
+*  along with GLPK. If not, see <http://www.gnu.org/licenses/>.
+***********************************************************************/
+
+#include "env.h"
+#include "npp.h"
+
+/***********************************************************************
+*  npp_sat_free_row - process free (unbounded) row
+*
+*  This routine processes row p, which is free (i.e. has no finite
+*  bounds):
+*
+*     -inf < sum a[p,j] x[j] < +inf.                                 (1)
+*
+*  The constraint (1) cannot be active and therefore it is redundant,
+*  so the routine simply removes it from the original problem. */
+
+void npp_sat_free_row(NPP *npp, NPPROW *p)
+{     /* the row should be free */
+      xassert(p->lb == -DBL_MAX && p->ub == +DBL_MAX);
+      /* remove the row from the problem */
+      npp_del_row(npp, p);
+      return;
+}
+
+/***********************************************************************
+*  npp_sat_fixed_col - process fixed column
+*
+*  This routine processes column q, which is fixed:
+*
+*     x[q] = s[q],                                                   (1)
+*
+*  where s[q] is a fixed column value.
+*
+*  The routine substitutes fixed value s[q] into constraint rows and
+*  then removes column x[q] from the original problem.
+*
+*  Substitution of x[q] = s[q] into row i gives:
+*
+*     L[i] <= sum a[i,j] x[j] <= U[i]   ==>
+*              j
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] x[q] <= U[i]   ==>
+*            j!=q
+*
+*     L[i] <= sum a[i,j] x[j] + a[i,q] s[q] <= U[i]   ==>
+*            j!=q
+*
+*     L~[i] <= sum a[i,j] x[j] <= U~[i],
+*             j!=q
+*
+*  where
+*
+*     L~[i] = L[i] - a[i,q] s[q],                                    (2)
+*
+*     U~[i] = U[i] - a[i,q] s[q]                                     (3)
+*
+*  are, respectively, lower and upper bound of row i in the transformed
+*  problem.
+*
+*  On recovering solution x[q] is assigned the value of s[q]. */
+
+struct sat_fixed_col
+{     /* fixed column */
+      int q;
+      /* column reference number for variable x[q] */
+      int s;
+      /* value, at which x[q] is fixed */
+};
+
+static int rcv_sat_fixed_col(NPP *, void *);
+
+int npp_sat_fixed_col(NPP *npp, NPPCOL *q)
+{     struct sat_fixed_col *info;
+      NPPROW *i;
+      NPPAIJ *aij;
+      int temp;
+      /* the column should be fixed */
+      xassert(q->lb == q->ub);
+      /* create transformation stack entry */
+      info = npp_push_tse(npp,
+         rcv_sat_fixed_col, sizeof(struct sat_fixed_col));
+      info->q = q->j;
+      info->s = (int)q->lb;
+      xassert((double)info->s == q->lb);
+      /* substitute x[q] = s[q] into constraint rows */
+      if (info->s == 0)
+         goto skip;
+      for (aij = q->ptr; aij != NULL; aij = aij->c_next)
+      {  i = aij->row;
+         if (i->lb != -DBL_MAX)
+         {  i->lb -= aij->val * (double)info->s;
+            temp = (int)i->lb;
+            if ((double)temp != i->lb)
+               return 1; /* integer arithmetic error */
+         }
+         if (i->ub != +DBL_MAX)
+         {  i->ub -= aij->val * (double)info->s;
+            temp = (int)i->ub;
+            if ((double)temp != i->ub)
+               return 2; /* integer arithmetic error */
+         }
+      }
+skip: /* remove the column from the problem */
+      npp_del_col(npp, q);
+      return 0;
+}
+
+static int rcv_sat_fixed_col(NPP *npp, void *info_)
+{     struct sat_fixed_col *info = info_;
+      npp->c_value[info->q] = (double)info->s;
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_is_bin_comb - test if row is binary combination
+*
+*  This routine tests if the specified row is a binary combination,
+*  i.e. all its constraint coefficients are +1 and -1 and all variables
+*  are binary. If the test was passed, the routine returns non-zero,
+*  otherwise zero. */
+
+int npp_sat_is_bin_comb(NPP *npp, NPPROW *row)
+{     NPPCOL *col;
+      NPPAIJ *aij;
+      xassert(npp == npp);
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  if (!(aij->val == +1.0 || aij->val == -1.0))
+            return 0; /* non-unity coefficient */
+         col = aij->col;
+         if (!(col->is_int && col->lb == 0.0 && col->ub == 1.0))
+            return 0; /* non-binary column */
+      }
+      return 1; /* test was passed */
+}
+
+/***********************************************************************
+*  npp_sat_num_pos_coef - determine number of positive coefficients
+*
+*  This routine returns the number of positive coefficients in the
+*  specified row. */
+
+int npp_sat_num_pos_coef(NPP *npp, NPPROW *row)
+{     NPPAIJ *aij;
+      int num = 0;
+      xassert(npp == npp);
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->val > 0.0)
+            num++;
+      }
+      return num;
+}
+
+/***********************************************************************
+*  npp_sat_num_neg_coef - determine number of negative coefficients
+*
+*  This routine returns the number of negative coefficients in the
+*  specified row. */
+
+int npp_sat_num_neg_coef(NPP *npp, NPPROW *row)
+{     NPPAIJ *aij;
+      int num = 0;
+      xassert(npp == npp);
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->val < 0.0)
+            num++;
+      }
+      return num;
+}
+
+/***********************************************************************
+*  npp_sat_is_cover_ineq - test if row is covering inequality
+*
+*  The canonical form of a covering inequality is the following:
+*
+*     sum x[j] >= 1,                                                 (1)
+*   j in J
+*
+*  where all x[j] are binary variables.
+*
+*  In general case a covering inequality may have one of the following
+*  two forms:
+*
+*     sum  x[j] -  sum  x[j] >= 1 - |J-|,                            (2)
+*   j in J+      j in J-
+*
+*
+*     sum  x[j] -  sum  x[j] <= |J+| - 1.                            (3)
+*   j in J+      j in J-
+*
+*  Obviously, the inequality (2) can be transformed to the form (1) by
+*  substitution x[j] = 1 - x'[j] for all j in J-, where x'[j] is the
+*  negation of variable x[j]. And the inequality (3) can be transformed
+*  to (2) by multiplying both left- and right-hand sides by -1.
+*
+*  This routine returns one of the following codes:
+*
+*  0, if the specified row is not a covering inequality;
+*
+*  1, if the specified row has the form (2);
+*
+*  2, if the specified row has the form (3). */
+
+int npp_sat_is_cover_ineq(NPP *npp, NPPROW *row)
+{     xassert(npp == npp);
+      if (row->lb != -DBL_MAX && row->ub == +DBL_MAX)
+      {  /* row is inequality of '>=' type */
+         if (npp_sat_is_bin_comb(npp, row))
+         {  /* row is a binary combination */
+            if (row->lb == 1.0 - npp_sat_num_neg_coef(npp, row))
+            {  /* row has the form (2) */
+               return 1;
+            }
+         }
+      }
+      else if (row->lb == -DBL_MAX && row->ub != +DBL_MAX)
+      {  /* row is inequality of '<=' type */
+         if (npp_sat_is_bin_comb(npp, row))
+         {  /* row is a binary combination */
+            if (row->ub == npp_sat_num_pos_coef(npp, row) - 1.0)
+            {  /* row has the form (3) */
+               return 2;
+            }
+         }
+      }
+      /* row is not a covering inequality */
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_is_pack_ineq - test if row is packing inequality
+*
+*  The canonical form of a packing inequality is the following:
+*
+*     sum x[j] <= 1,                                                 (1)
+*   j in J
+*
+*  where all x[j] are binary variables.
+*
+*  In general case a packing inequality may have one of the following
+*  two forms:
+*
+*     sum  x[j] -  sum  x[j] <= 1 - |J-|,                            (2)
+*   j in J+      j in J-
+*
+*
+*     sum  x[j] -  sum  x[j] >= |J+| - 1.                            (3)
+*   j in J+      j in J-
+*
+*  Obviously, the inequality (2) can be transformed to the form (1) by
+*  substitution x[j] = 1 - x'[j] for all j in J-, where x'[j] is the
+*  negation of variable x[j]. And the inequality (3) can be transformed
+*  to (2) by multiplying both left- and right-hand sides by -1.
+*
+*  This routine returns one of the following codes:
+*
+*  0, if the specified row is not a packing inequality;
+*
+*  1, if the specified row has the form (2);
+*
+*  2, if the specified row has the form (3). */
+
+int npp_sat_is_pack_ineq(NPP *npp, NPPROW *row)
+{     xassert(npp == npp);
+      if (row->lb == -DBL_MAX && row->ub != +DBL_MAX)
+      {  /* row is inequality of '<=' type */
+         if (npp_sat_is_bin_comb(npp, row))
+         {  /* row is a binary combination */
+            if (row->ub == 1.0 - npp_sat_num_neg_coef(npp, row))
+            {  /* row has the form (2) */
+               return 1;
+            }
+         }
+      }
+      else if (row->lb != -DBL_MAX && row->ub == +DBL_MAX)
+      {  /* row is inequality of '>=' type */
+         if (npp_sat_is_bin_comb(npp, row))
+         {  /* row is a binary combination */
+            if (row->lb == npp_sat_num_pos_coef(npp, row) - 1.0)
+            {  /* row has the form (3) */
+               return 2;
+            }
+         }
+      }
+      /* row is not a packing inequality */
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_is_partn_eq - test if row is partitioning equality
+*
+*  The canonical form of a partitioning equality is the following:
+*
+*     sum x[j] = 1,                                                  (1)
+*   j in J
+*
+*  where all x[j] are binary variables.
+*
+*  In general case a partitioning equality may have one of the following
+*  two forms:
+*
+*     sum  x[j] -  sum  x[j] = 1 - |J-|,                             (2)
+*   j in J+      j in J-
+*
+*
+*     sum  x[j] -  sum  x[j] = |J+| - 1.                             (3)
+*   j in J+      j in J-
+*
+*  Obviously, the equality (2) can be transformed to the form (1) by
+*  substitution x[j] = 1 - x'[j] for all j in J-, where x'[j] is the
+*  negation of variable x[j]. And the equality (3) can be transformed
+*  to (2) by multiplying both left- and right-hand sides by -1.
+*
+*  This routine returns one of the following codes:
+*
+*  0, if the specified row is not a partitioning equality;
+*
+*  1, if the specified row has the form (2);
+*
+*  2, if the specified row has the form (3). */
+
+int npp_sat_is_partn_eq(NPP *npp, NPPROW *row)
+{     xassert(npp == npp);
+      if (row->lb == row->ub)
+      {  /* row is equality constraint */
+         if (npp_sat_is_bin_comb(npp, row))
+         {  /* row is a binary combination */
+            if (row->lb == 1.0 - npp_sat_num_neg_coef(npp, row))
+            {  /* row has the form (2) */
+               return 1;
+            }
+            if (row->ub == npp_sat_num_pos_coef(npp, row) - 1.0)
+            {  /* row has the form (3) */
+               return 2;
+            }
+         }
+      }
+      /* row is not a partitioning equality */
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_reverse_row - multiply both sides of row by -1
+*
+*  This routines multiplies by -1 both left- and right-hand sides of
+*  the specified row:
+*
+*     L <= sum x[j] <= U,
+*
+*  that results in the following row:
+*
+*     -U <= sum (-x[j]) <= -L.
+*
+*  If no integer overflow occured, the routine returns zero, otherwise
+*  non-zero. */
+
+int npp_sat_reverse_row(NPP *npp, NPPROW *row)
+{     NPPAIJ *aij;
+      int temp, ret = 0;
+      double old_lb, old_ub;
+      xassert(npp == npp);
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  aij->val = -aij->val;
+         temp = (int)aij->val;
+         if ((double)temp != aij->val)
+            ret = 1;
+      }
+      old_lb = row->lb, old_ub = row->ub;
+      if (old_ub == +DBL_MAX)
+         row->lb = -DBL_MAX;
+      else
+      {  row->lb = -old_ub;
+         temp = (int)row->lb;
+         if ((double)temp != row->lb)
+            ret = 2;
+      }
+      if (old_lb == -DBL_MAX)
+         row->ub = +DBL_MAX;
+      else
+      {  row->ub = -old_lb;
+         temp = (int)row->ub;
+         if ((double)temp != row->ub)
+            ret = 3;
+      }
+      return ret;
+}
+
+/***********************************************************************
+*  npp_sat_split_pack - split packing inequality
+*
+*  Let there be given a packing inequality in canonical form:
+*
+*     sum  t[j] <= 1,                                                (1)
+*   j in J
+*
+*  where t[j] = x[j] or t[j] = 1 - x[j], x[j] is a binary variable.
+*  And let J = J1 U J2 is a partition of the set of literals. Then the
+*  inequality (1) is obviously equivalent to the following two packing
+*  inequalities:
+*
+*     sum  t[j] <= y       <-->   sum  t[j] + (1 - y) <= 1,          (2)
+*   j in J1                     j in J1
+*
+*     sum  t[j] <= 1 - y   <-->   sum  t[j] + y       <= 1,          (3)
+*   j in J2                     j in J2
+*
+*  where y is a new binary variable added to the transformed problem.
+*
+*  Assuming that the specified row is a packing inequality (1), this
+*  routine constructs the set J1 by including there first nlit literals
+*  (terms) from the specified row, and the set J2 = J \ J1. Then the
+*  routine creates a new row, which corresponds to inequality (2), and
+*  replaces the specified row with inequality (3). */
+
+NPPROW *npp_sat_split_pack(NPP *npp, NPPROW *row, int nlit)
+{     NPPROW *rrr;
+      NPPCOL *col;
+      NPPAIJ *aij;
+      int k;
+      /* original row should be packing inequality (1) */
+      xassert(npp_sat_is_pack_ineq(npp, row) == 1);
+      /* and nlit should be less than the number of literals (terms)
+         in the original row */
+      xassert(0 < nlit && nlit < npp_row_nnz(npp, row));
+      /* create new row corresponding to inequality (2) */
+      rrr = npp_add_row(npp);
+      rrr->lb = -DBL_MAX, rrr->ub = 1.0;
+      /* move first nlit literals (terms) from the original row to the
+         new row; the original row becomes inequality (3) */
+      for (k = 1; k <= nlit; k++)
+      {  aij = row->ptr;
+         xassert(aij != NULL);
+         /* add literal to the new row */
+         npp_add_aij(npp, rrr, aij->col, aij->val);
+         /* correct rhs */
+         if (aij->val < 0.0)
+            rrr->ub -= 1.0, row->ub += 1.0;
+         /* remove literal from the original row */
+         npp_del_aij(npp, aij);
+      }
+      /* create new binary variable y */
+      col = npp_add_col(npp);
+      col->is_int = 1, col->lb = 0.0, col->ub = 1.0;
+      /* include literal (1 - y) in the new row */
+      npp_add_aij(npp, rrr, col, -1.0);
+      rrr->ub -= 1.0;
+      /* include literal y in the original row */
+      npp_add_aij(npp, row, col, +1.0);
+      return rrr;
+}
+
+/***********************************************************************
+*  npp_sat_encode_pack - encode packing inequality
+*
+*  Given a packing inequality in canonical form:
+*
+*     sum  t[j] <= 1,                                                (1)
+*   j in J
+*
+*  where t[j] = x[j] or t[j] = 1 - x[j], x[j] is a binary variable,
+*  this routine translates it to CNF by replacing it with the following
+*  equivalent set of edge packing inequalities:
+*
+*     t[j] + t[k] <= 1   for all j, k in J, j != k.                  (2)
+*
+*  Then the routine transforms each edge packing inequality (2) to
+*  corresponding covering inequality (that encodes two-literal clause)
+*  by multiplying both its part by -1:
+*
+*     - t[j] - t[k] >= -1   <-->   (1 - t[j]) + (1 - t[k]) >= 1.     (3)
+*
+*  On exit the routine removes the original row from the problem. */
+
+void npp_sat_encode_pack(NPP *npp, NPPROW *row)
+{     NPPROW *rrr;
+      NPPAIJ *aij, *aik;
+      /* original row should be packing inequality (1) */
+      xassert(npp_sat_is_pack_ineq(npp, row) == 1);
+      /* create equivalent system of covering inequalities (3) */
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  /* due to symmetry only one of inequalities t[j] + t[k] <= 1
+            and t[k] <= t[j] <= 1 can be considered */
+         for (aik = aij->r_next; aik != NULL; aik = aik->r_next)
+         {  /* create edge packing inequality (2) */
+            rrr = npp_add_row(npp);
+            rrr->lb = -DBL_MAX, rrr->ub = 1.0;
+            npp_add_aij(npp, rrr, aij->col, aij->val);
+            if (aij->val < 0.0)
+               rrr->ub -= 1.0;
+            npp_add_aij(npp, rrr, aik->col, aik->val);
+            if (aik->val < 0.0)
+               rrr->ub -= 1.0;
+            /* and transform it to covering inequality (3) */
+            npp_sat_reverse_row(npp, rrr);
+            xassert(npp_sat_is_cover_ineq(npp, rrr) == 1);
+         }
+      }
+      /* remove the original row from the problem */
+      npp_del_row(npp, row);
+      return;
+}
+
+/***********************************************************************
+*  npp_sat_encode_sum2 - encode 2-bit summation
+*
+*  Given a set containing two literals x and y this routine encodes
+*  the equality
+*
+*     x + y = s + 2 * c,                                             (1)
+*
+*  where
+*
+*     s = (x + y) % 2                                                (2)
+*
+*  is a binary variable modeling the low sum bit, and
+*
+*     c = (x + y) / 2                                                (3)
+*
+*  is a binary variable modeling the high (carry) sum bit. */
+
+void npp_sat_encode_sum2(NPP *npp, NPPLSE *set, NPPSED *sed)
+{     NPPROW *row;
+      int x, y, s, c;
+      /* the set should contain exactly two literals */
+      xassert(set != NULL);
+      xassert(set->next != NULL);
+      xassert(set->next->next == NULL);
+      sed->x = set->lit;
+      xassert(sed->x.neg == 0 || sed->x.neg == 1);
+      sed->y = set->next->lit;
+      xassert(sed->y.neg == 0 || sed->y.neg == 1);
+      sed->z.col = NULL, sed->z.neg = 0;
+      /* perform encoding s = (x + y) % 2 */
+      sed->s = npp_add_col(npp);
+      sed->s->is_int = 1, sed->s->lb = 0.0, sed->s->ub = 1.0;
+      for (x = 0; x <= 1; x++)
+      {  for (y = 0; y <= 1; y++)
+         {  for (s = 0; s <= 1; s++)
+            {  if ((x + y) % 2 != s)
+               {  /* generate CNF clause to disable infeasible
+                     combination */
+                  row = npp_add_row(npp);
+                  row->lb = 1.0, row->ub = +DBL_MAX;
+                  if (x == sed->x.neg)
+                     npp_add_aij(npp, row, sed->x.col, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->x.col, -1.0);
+                     row->lb -= 1.0;
+                  }
+                  if (y == sed->y.neg)
+                     npp_add_aij(npp, row, sed->y.col, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->y.col, -1.0);
+                     row->lb -= 1.0;
+                  }
+                  if (s == 0)
+                     npp_add_aij(npp, row, sed->s, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->s, -1.0);
+                     row->lb -= 1.0;
+                  }
+               }
+            }
+         }
+      }
+      /* perform encoding c = (x + y) / 2 */
+      sed->c = npp_add_col(npp);
+      sed->c->is_int = 1, sed->c->lb = 0.0, sed->c->ub = 1.0;
+      for (x = 0; x <= 1; x++)
+      {  for (y = 0; y <= 1; y++)
+         {  for (c = 0; c <= 1; c++)
+            {  if ((x + y) / 2 != c)
+               {  /* generate CNF clause to disable infeasible
+                     combination */
+                  row = npp_add_row(npp);
+                  row->lb = 1.0, row->ub = +DBL_MAX;
+                  if (x == sed->x.neg)
+                     npp_add_aij(npp, row, sed->x.col, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->x.col, -1.0);
+                     row->lb -= 1.0;
+                  }
+                  if (y == sed->y.neg)
+                     npp_add_aij(npp, row, sed->y.col, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->y.col, -1.0);
+                     row->lb -= 1.0;
+                  }
+                  if (c == 0)
+                     npp_add_aij(npp, row, sed->c, +1.0);
+                  else
+                  {  npp_add_aij(npp, row, sed->c, -1.0);
+                     row->lb -= 1.0;
+                  }
+               }
+            }
+         }
+      }
+      return;
+}
+
+/***********************************************************************
+*  npp_sat_encode_sum3 - encode 3-bit summation
+*
+*  Given a set containing at least three literals this routine chooses
+*  some literals x, y, z from that set and encodes the equality
+*
+*     x + y + z = s + 2 * c,                                         (1)
+*
+*  where
+*
+*     s = (x + y + z) % 2                                            (2)
+*
+*  is a binary variable modeling the low sum bit, and
+*
+*     c = (x + y + z) / 2                                            (3)
+*
+*  is a binary variable modeling the high (carry) sum bit. */
+
+void npp_sat_encode_sum3(NPP *npp, NPPLSE *set, NPPSED *sed)
+{     NPPROW *row;
+      int x, y, z, s, c;
+      /* the set should contain at least three literals */
+      xassert(set != NULL);
+      xassert(set->next != NULL);
+      xassert(set->next->next != NULL);
+      sed->x = set->lit;
+      xassert(sed->x.neg == 0 || sed->x.neg == 1);
+      sed->y = set->next->lit;
+      xassert(sed->y.neg == 0 || sed->y.neg == 1);
+      sed->z = set->next->next->lit;
+      xassert(sed->z.neg == 0 || sed->z.neg == 1);
+      /* perform encoding s = (x + y + z) % 2 */
+      sed->s = npp_add_col(npp);
+      sed->s->is_int = 1, sed->s->lb = 0.0, sed->s->ub = 1.0;
+      for (x = 0; x <= 1; x++)
+      {  for (y = 0; y <= 1; y++)
+         {  for (z = 0; z <= 1; z++)
+            {  for (s = 0; s <= 1; s++)
+               {  if ((x + y + z) % 2 != s)
+                  {  /* generate CNF clause to disable infeasible
+                        combination */
+                     row = npp_add_row(npp);
+                     row->lb = 1.0, row->ub = +DBL_MAX;
+                     if (x == sed->x.neg)
+                        npp_add_aij(npp, row, sed->x.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->x.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (y == sed->y.neg)
+                        npp_add_aij(npp, row, sed->y.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->y.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (z == sed->z.neg)
+                        npp_add_aij(npp, row, sed->z.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->z.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (s == 0)
+                        npp_add_aij(npp, row, sed->s, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->s, -1.0);
+                        row->lb -= 1.0;
+                     }
+                  }
+               }
+            }
+         }
+      }
+      /* perform encoding c = (x + y + z) / 2 */
+      sed->c = npp_add_col(npp);
+      sed->c->is_int = 1, sed->c->lb = 0.0, sed->c->ub = 1.0;
+      for (x = 0; x <= 1; x++)
+      {  for (y = 0; y <= 1; y++)
+         {  for (z = 0; z <= 1; z++)
+            {  for (c = 0; c <= 1; c++)
+               {  if ((x + y + z) / 2 != c)
+                  {  /* generate CNF clause to disable infeasible
+                        combination */
+                     row = npp_add_row(npp);
+                     row->lb = 1.0, row->ub = +DBL_MAX;
+                     if (x == sed->x.neg)
+                        npp_add_aij(npp, row, sed->x.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->x.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (y == sed->y.neg)
+                        npp_add_aij(npp, row, sed->y.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->y.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (z == sed->z.neg)
+                        npp_add_aij(npp, row, sed->z.col, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->z.col, -1.0);
+                        row->lb -= 1.0;
+                     }
+                     if (c == 0)
+                        npp_add_aij(npp, row, sed->c, +1.0);
+                     else
+                     {  npp_add_aij(npp, row, sed->c, -1.0);
+                        row->lb -= 1.0;
+                     }
+                  }
+               }
+            }
+         }
+      }
+      return;
+}
+
+/***********************************************************************
+*  npp_sat_encode_sum_ax - encode linear combination of 0-1 variables
+*
+*  PURPOSE
+*
+*  Given a linear combination of binary variables:
+*
+*     sum a[j] x[j],                                                 (1)
+*      j
+*
+*  which is the linear form of the specified row, this routine encodes
+*  (i.e. translates to CNF) the following equality:
+*
+*                        n
+*     sum |a[j]| t[j] = sum 2**(k-1) * y[k],                         (2)
+*      j                k=1
+*
+*  where t[j] = x[j] (if a[j] > 0) or t[j] = 1 - x[j] (if a[j] < 0),
+*  and y[k] is either t[j] or a new literal created by the routine or
+*  a constant zero. Note that the sum in the right-hand side of (2) can
+*  be thought as a n-bit representation of the sum in the left-hand
+*  side, which is a non-negative integer number.
+*
+*  ALGORITHM
+*
+*  First, the number of bits, n, sufficient to represent any value in
+*  the left-hand side of (2) is determined. Obviously, n is the number
+*  of bits sufficient to represent the sum (sum |a[j]|).
+*
+*  Let
+*
+*               n
+*     |a[j]| = sum 2**(k-1) b[j,k],                                  (3)
+*              k=1
+*
+*  where b[j,k] is k-th bit in a n-bit representation of |a[j]|. Then
+*
+*                          m            n
+*     sum |a[j]| * t[j] = sum 2**(k-1) sum b[j,k] * t[j].            (4)
+*      j                  k=1          j=1
+*
+*  Introducing the set
+*
+*     J[k] = { j : b[j,k] = 1 }                                      (5)
+*
+*  allows rewriting (4) as follows:
+*
+*                          n
+*     sum |a[j]| * t[j] = sum 2**(k-1)  sum    t[j].                 (6)
+*      j                  k=1         j in J[k]
+*
+*  Thus, our goal is to provide |J[k]| <= 1 for all k, in which case
+*  we will have the representation (1).
+*
+*  Let |J[k]| = 2, i.e. J[k] has exactly two literals u and v. In this
+*  case we can apply the following transformation:
+*
+*     u + v = s + 2 * c,                                             (7)
+*
+*  where s and c are, respectively, low (sum) and high (carry) bits of
+*  the sum of two bits. This allows to replace two literals u and v in
+*  J[k] by one literal s, and carry out literal c to J[k+1].
+*
+*  If |J[k]| >= 3, i.e. J[k] has at least three literals u, v, and w,
+*  we can apply the following transformation:
+*
+*     u + v + w = s + 2 * c.                                         (8)
+*
+*  Again, literal s replaces literals u, v, and w in J[k], and literal
+*  c goes into J[k+1].
+*
+*  On exit the routine stores each literal from J[k] in element y[k],
+*  1 <= k <= n. If J[k] is empty, y[k] is set to constant false.
+*
+*  RETURNS
+*
+*  The routine returns n, the number of literals in the right-hand side
+*  of (2), 0 <= n <= NBIT_MAX. If the sum (sum |a[j]|) is too large, so
+*  more than NBIT_MAX (= 31) literals are needed to encode the original
+*  linear combination, the routine returns a negative value. */
+
+#define NBIT_MAX 31
+/* maximal number of literals in the right hand-side of (2) */
+
+static NPPLSE *remove_lse(NPP *npp, NPPLSE *set, NPPCOL *col)
+{     /* remove specified literal from specified literal set */
+      NPPLSE *lse, *prev = NULL;
+      for (lse = set; lse != NULL; prev = lse, lse = lse->next)
+         if (lse->lit.col == col) break;
+      xassert(lse != NULL);
+      if (prev == NULL)
+         set = lse->next;
+      else
+         prev->next = lse->next;
+      dmp_free_atom(npp->pool, lse, sizeof(NPPLSE));
+      return set;
+}
+
+int npp_sat_encode_sum_ax(NPP *npp, NPPROW *row, NPPLIT y[])
+{     NPPAIJ *aij;
+      NPPLSE *set[1+NBIT_MAX], *lse;
+      NPPSED sed;
+      int k, n, temp;
+      double sum;
+      /* compute the sum (sum |a[j]|) */
+      sum = 0.0;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+         sum += fabs(aij->val);
+      /* determine n, the number of bits in the sum */
+      temp = (int)sum;
+      if ((double)temp != sum)
+         return -1; /* integer arithmetic error */
+      for (n = 0; temp > 0; n++, temp >>= 1);
+      xassert(0 <= n && n <= NBIT_MAX);
+      /* build initial sets J[k], 1 <= k <= n; see (5) */
+      /* set[k] is a pointer to the list of literals in J[k] */
+      for (k = 1; k <= n; k++)
+         set[k] = NULL;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  temp = (int)fabs(aij->val);
+         xassert((int)temp == fabs(aij->val));
+         for (k = 1; temp > 0; k++, temp >>= 1)
+         {  if (temp & 1)
+            {  xassert(k <= n);
+               lse = dmp_get_atom(npp->pool, sizeof(NPPLSE));
+               lse->lit.col = aij->col;
+               lse->lit.neg = (aij->val > 0.0 ? 0 : 1);
+               lse->next = set[k];
+               set[k] = lse;
+            }
+         }
+      }
+      /* main transformation loop */
+      for (k = 1; k <= n; k++)
+      {  /* reduce J[k] and set y[k] */
+         for (;;)
+         {  if (set[k] == NULL)
+            {  /* J[k] is empty */
+               /* set y[k] to constant false */
+               y[k].col = NULL, y[k].neg = 0;
+               break;
+            }
+            if (set[k]->next == NULL)
+            {  /* J[k] contains one literal */
+               /* set y[k] to that literal */
+               y[k] = set[k]->lit;
+               dmp_free_atom(npp->pool, set[k], sizeof(NPPLSE));
+               break;
+            }
+            if (set[k]->next->next == NULL)
+            {  /* J[k] contains two literals */
+               /* apply transformation (7) */
+               npp_sat_encode_sum2(npp, set[k], &sed);
+            }
+            else
+            {  /* J[k] contains at least three literals */
+               /* apply transformation (8) */
+               npp_sat_encode_sum3(npp, set[k], &sed);
+               /* remove third literal from set[k] */
+               set[k] = remove_lse(npp, set[k], sed.z.col);
+            }
+            /* remove second literal from set[k] */
+            set[k] = remove_lse(npp, set[k], sed.y.col);
+            /* remove first literal from set[k] */
+            set[k] = remove_lse(npp, set[k], sed.x.col);
+            /* include new literal s to set[k] */
+            lse = dmp_get_atom(npp->pool, sizeof(NPPLSE));
+            lse->lit.col = sed.s, lse->lit.neg = 0;
+            lse->next = set[k];
+            set[k] = lse;
+            /* include new literal c to set[k+1] */
+            xassert(k < n); /* FIXME: can "overflow" happen? */
+            lse = dmp_get_atom(npp->pool, sizeof(NPPLSE));
+            lse->lit.col = sed.c, lse->lit.neg = 0;
+            lse->next = set[k+1];
+            set[k+1] = lse;
+         }
+      }
+      return n;
+}
+
+/***********************************************************************
+*  npp_sat_normalize_clause - normalize clause
+*
+*  This routine normalizes the specified clause, which is a disjunction
+*  of literals, by replacing multiple literals, which refer to the same
+*  binary variable, with a single literal.
+*
+*  On exit the routine returns the number of literals in the resulting
+*  clause. However, if the specified clause includes both a literal and
+*  its negation, the routine returns a negative value meaning that the
+*  clause is equivalent to the value true. */
+
+int npp_sat_normalize_clause(NPP *npp, int size, NPPLIT lit[])
+{     int j, k, new_size;
+      xassert(npp == npp);
+      xassert(size >= 0);
+      new_size = 0;
+      for (k = 1; k <= size; k++)
+      {  for (j = 1; j <= new_size; j++)
+         {  if (lit[k].col == lit[j].col)
+            {  /* lit[k] refers to the same variable as lit[j], which
+                  is already included in the resulting clause */
+               if (lit[k].neg == lit[j].neg)
+               {  /* ignore lit[k] due to the idempotent law */
+                  goto skip;
+               }
+               else
+               {  /* lit[k] is NOT lit[j]; the clause is equivalent to
+                     the value true */
+                  return -1;
+               }
+            }
+         }
+         /* include lit[k] in the resulting clause */
+         lit[++new_size] = lit[k];
+skip:    ;
+      }
+      return new_size;
+}
+
+/***********************************************************************
+*  npp_sat_encode_clause - translate clause to cover inequality
+*
+*  Given a clause
+*
+*     OR  t[j],                                                      (1)
+*   j in J
+*
+*  where t[j] is a literal, i.e. t[j] = x[j] or t[j] = NOT x[j], this
+*  routine translates it to the following equivalent cover inequality,
+*  which is added to the transformed problem:
+*
+*     sum t[j] >= 1,                                                 (2)
+*   j in J
+*
+*  where t[j] = x[j] or t[j] = 1 - x[j].
+*
+*  If necessary, the clause should be normalized before a call to this
+*  routine. */
+
+NPPROW *npp_sat_encode_clause(NPP *npp, int size, NPPLIT lit[])
+{     NPPROW *row;
+      int k;
+      xassert(size >= 1);
+      row = npp_add_row(npp);
+      row->lb = 1.0, row->ub = +DBL_MAX;
+      for (k = 1; k <= size; k++)
+      {  xassert(lit[k].col != NULL);
+         if (lit[k].neg == 0)
+            npp_add_aij(npp, row, lit[k].col, +1.0);
+         else if (lit[k].neg == 1)
+         {  npp_add_aij(npp, row, lit[k].col, -1.0);
+            row->lb -= 1.0;
+         }
+         else
+            xassert(lit != lit);
+      }
+      return row;
+}
+
+/***********************************************************************
+*  npp_sat_encode_geq - encode "not less than" constraint
+*
+*  PURPOSE
+*
+*  This routine translates to CNF the following constraint:
+*
+*      n
+*     sum 2**(k-1) * y[k] >= b,                                      (1)
+*     k=1
+*
+*  where y[k] is either a literal (i.e. y[k] = x[k] or y[k] = 1 - x[k])
+*  or constant false (zero), b is a given lower bound.
+*
+*  ALGORITHM
+*
+*  If b < 0, the constraint is redundant, so assume that b >= 0. Let
+*
+*          n
+*     b = sum 2**(k-1) b[k],                                         (2)
+*         k=1
+*
+*  where b[k] is k-th binary digit of b. (Note that if b >= 2**n and
+*  therefore cannot be represented in the form (2), the constraint (1)
+*  is infeasible.) In this case the condition (1) is equivalent to the
+*  following condition:
+*
+*     y[n] y[n-1] ... y[2] y[1] >= b[n] b[n-1] ... b[2] b[1],        (3)
+*
+*  where ">=" is understood lexicographically.
+*
+*  Algorithmically the condition (3) can be tested as follows:
+*
+*     for (k = n; k >= 1; k--)
+*     {  if (y[k] < b[k])
+*           y is less than b;
+*        if (y[k] > b[k])
+*           y is greater than b;
+*     }
+*     y is equal to b;
+*
+*  Thus, y is less than b iff there exists k, 1 <= k <= n, for which
+*  the following condition is satisfied:
+*
+*     y[n] = b[n] AND ... AND y[k+1] = b[k+1] AND y[k] < b[k].       (4)
+*
+*  Negating the condition (4) we have that y is not less than b iff for
+*  all k, 1 <= k <= n, the following condition is satisfied:
+*
+*     y[n] != b[n] OR ... OR y[k+1] != b[k+1] OR y[k] >= b[k].       (5)
+*
+*  Note that if b[k] = 0, the literal y[k] >= b[k] is always true, in
+*  which case the entire clause (5) is true and can be omitted.
+*
+*  RETURNS
+*
+*  Normally the routine returns zero. However, if the constraint (1) is
+*  infeasible, the routine returns non-zero. */
+
+int npp_sat_encode_geq(NPP *npp, int n, NPPLIT y[], int rhs)
+{     NPPLIT lit[1+NBIT_MAX];
+      int j, k, size, temp, b[1+NBIT_MAX];
+      xassert(0 <= n && n <= NBIT_MAX);
+      /* if the constraint (1) is redundant, do nothing */
+      if (rhs < 0)
+         return 0;
+      /* determine binary digits of b according to (2) */
+      for (k = 1, temp = rhs; k <= n; k++, temp >>= 1)
+         b[k] = temp & 1;
+      if (temp != 0)
+      {  /* b >= 2**n; the constraint (1) is infeasible */
+         return 1;
+      }
+      /* main transformation loop */
+      for (k = 1; k <= n; k++)
+      {  /* build the clause (5) for current k */
+         size = 0; /* clause size = number of literals */
+         /* add literal y[k] >= b[k] */
+         if (b[k] == 0)
+         {  /* b[k] = 0 -> the literal is true */
+            goto skip;
+         }
+         else if (y[k].col == NULL)
+         {  /* y[k] = 0, b[k] = 1 -> the literal is false */
+            xassert(y[k].neg == 0);
+         }
+         else
+         {  /* add literal y[k] = 1 */
+            lit[++size] = y[k];
+         }
+         for (j = k+1; j <= n; j++)
+         {  /* add literal y[j] != b[j] */
+            if (y[j].col == NULL)
+            {  xassert(y[j].neg == 0);
+               if (b[j] == 0)
+               {  /* y[j] = 0, b[j] = 0 -> the literal is false */
+                  continue;
+               }
+               else
+               {  /* y[j] = 0, b[j] = 1 -> the literal is true */
+                  goto skip;
+               }
+            }
+            else
+            {  lit[++size] = y[j];
+               if (b[j] != 0)
+                  lit[size].neg = 1 - lit[size].neg;
+            }
+         }
+         /* normalize the clause */
+         size = npp_sat_normalize_clause(npp, size, lit);
+         if (size < 0)
+         {  /* the clause is equivalent to the value true */
+            goto skip;
+         }
+         if (size == 0)
+         {  /* the clause is equivalent to the value false; this means
+               that the constraint (1) is infeasible */
+            return 2;
+         }
+         /* translate the clause to corresponding cover inequality */
+         npp_sat_encode_clause(npp, size, lit);
+skip:    ;
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_encode_leq - encode "not greater than" constraint
+*
+*  PURPOSE
+*
+*  This routine translates to CNF the following constraint:
+*
+*      n
+*     sum 2**(k-1) * y[k] <= b,                                      (1)
+*     k=1
+*
+*  where y[k] is either a literal (i.e. y[k] = x[k] or y[k] = 1 - x[k])
+*  or constant false (zero), b is a given upper bound.
+*
+*  ALGORITHM
+*
+*  If b < 0, the constraint is infeasible, so assume that b >= 0. Let
+*
+*          n
+*     b = sum 2**(k-1) b[k],                                         (2)
+*         k=1
+*
+*  where b[k] is k-th binary digit of b. (Note that if b >= 2**n and
+*  therefore cannot be represented in the form (2), the constraint (1)
+*  is redundant.) In this case the condition (1) is equivalent to the
+*  following condition:
+*
+*     y[n] y[n-1] ... y[2] y[1] <= b[n] b[n-1] ... b[2] b[1],        (3)
+*
+*  where "<=" is understood lexicographically.
+*
+*  Algorithmically the condition (3) can be tested as follows:
+*
+*     for (k = n; k >= 1; k--)
+*     {  if (y[k] < b[k])
+*           y is less than b;
+*        if (y[k] > b[k])
+*           y is greater than b;
+*     }
+*     y is equal to b;
+*
+*  Thus, y is greater than b iff there exists k, 1 <= k <= n, for which
+*  the following condition is satisfied:
+*
+*     y[n] = b[n] AND ... AND y[k+1] = b[k+1] AND y[k] > b[k].       (4)
+*
+*  Negating the condition (4) we have that y is not greater than b iff
+*  for all k, 1 <= k <= n, the following condition is satisfied:
+*
+*     y[n] != b[n] OR ... OR y[k+1] != b[k+1] OR y[k] <= b[k].       (5)
+*
+*  Note that if b[k] = 1, the literal y[k] <= b[k] is always true, in
+*  which case the entire clause (5) is true and can be omitted.
+*
+*  RETURNS
+*
+*  Normally the routine returns zero. However, if the constraint (1) is
+*  infeasible, the routine returns non-zero. */
+
+int npp_sat_encode_leq(NPP *npp, int n, NPPLIT y[], int rhs)
+{     NPPLIT lit[1+NBIT_MAX];
+      int j, k, size, temp, b[1+NBIT_MAX];
+      xassert(0 <= n && n <= NBIT_MAX);
+      /* check if the constraint (1) is infeasible */
+      if (rhs < 0)
+         return 1;
+      /* determine binary digits of b according to (2) */
+      for (k = 1, temp = rhs; k <= n; k++, temp >>= 1)
+         b[k] = temp & 1;
+      if (temp != 0)
+      {  /* b >= 2**n; the constraint (1) is redundant */
+         return 0;
+      }
+      /* main transformation loop */
+      for (k = 1; k <= n; k++)
+      {  /* build the clause (5) for current k */
+         size = 0; /* clause size = number of literals */
+         /* add literal y[k] <= b[k] */
+         if (b[k] == 1)
+         {  /* b[k] = 1 -> the literal is true */
+            goto skip;
+         }
+         else if (y[k].col == NULL)
+         {  /* y[k] = 0, b[k] = 0 -> the literal is true */
+            xassert(y[k].neg == 0);
+            goto skip;
+         }
+         else
+         {  /* add literal y[k] = 0 */
+            lit[++size] = y[k];
+            lit[size].neg = 1 - lit[size].neg;
+         }
+         for (j = k+1; j <= n; j++)
+         {  /* add literal y[j] != b[j] */
+            if (y[j].col == NULL)
+            {  xassert(y[j].neg == 0);
+               if (b[j] == 0)
+               {  /* y[j] = 0, b[j] = 0 -> the literal is false */
+                  continue;
+               }
+               else
+               {  /* y[j] = 0, b[j] = 1 -> the literal is true */
+                  goto skip;
+               }
+            }
+            else
+            {  lit[++size] = y[j];
+               if (b[j] != 0)
+                  lit[size].neg = 1 - lit[size].neg;
+            }
+         }
+         /* normalize the clause */
+         size = npp_sat_normalize_clause(npp, size, lit);
+         if (size < 0)
+         {  /* the clause is equivalent to the value true */
+            goto skip;
+         }
+         if (size == 0)
+         {  /* the clause is equivalent to the value false; this means
+               that the constraint (1) is infeasible */
+            return 2;
+         }
+         /* translate the clause to corresponding cover inequality */
+         npp_sat_encode_clause(npp, size, lit);
+skip:    ;
+      }
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_encode_row - encode constraint (row) of general type
+*
+*  PURPOSE
+*
+*  This routine translates to CNF the following constraint (row):
+*
+*     L <= sum a[j] x[j] <= U,                                       (1)
+*           j
+*
+*  where all x[j] are binary variables.
+*
+*  ALGORITHM
+*
+*  First, the routine performs substitution x[j] = t[j] for j in J+
+*  and x[j] = 1 - t[j] for j in J-, where J+ = { j : a[j] > 0 } and
+*  J- = { j : a[j] < 0 }. This gives:
+*
+*     L <=  sum  a[j] t[j] +   sum  a[j] (1 - t[j]) <= U  ==>
+*         j in J+            j in J-
+*
+*     L' <= sum |a[j]| t[j] <= U',                                   (2)
+*            j
+*
+*  where
+*
+*     L' = L -   sum  a[j],   U' = U -   sum  a[j].                  (3)
+*              j in J-                 j in J-
+*
+*  (Actually only new bounds L' and U' are computed.)
+*
+*  Then the routine translates to CNF the following equality:
+*
+*                        n
+*     sum |a[j]| t[j] = sum 2**(k-1) * y[k],                         (4)
+*      j                k=1
+*
+*  where y[k] is either some t[j] or a new literal or a constant zero
+*  (see the routine npp_sat_encode_sum_ax).
+*
+*  Finally, the routine translates to CNF the following conditions:
+*
+*      n
+*     sum 2**(k-1) * y[k] >= L'                                      (5)
+*     k=1
+*
+*  and
+*
+*      n
+*     sum 2**(k-1) * y[k] <= U'                                      (6)
+*     k=1
+*
+*  (see the routines npp_sat_encode_geq and npp_sat_encode_leq).
+*
+*  All resulting clauses are encoded as cover inequalities and included
+*  into the transformed problem.
+*
+*  Note that on exit the routine removes the specified constraint (row)
+*  from the original problem.
+*
+*  RETURNS
+*
+*  The routine returns one of the following codes:
+*
+*  0 - translation was successful;
+*  1 - constraint (1) was found infeasible;
+*  2 - integer arithmetic error occured. */
+
+int npp_sat_encode_row(NPP *npp, NPPROW *row)
+{     NPPAIJ *aij;
+      NPPLIT y[1+NBIT_MAX];
+      int n, rhs;
+      double lb, ub;
+      /* the row should not be free */
+      xassert(!(row->lb == -DBL_MAX && row->ub == +DBL_MAX));
+      /* compute new bounds L' and U' (3) */
+      lb = row->lb;
+      ub = row->ub;
+      for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+      {  if (aij->val < 0.0)
+         {  if (lb != -DBL_MAX)
+               lb -= aij->val;
+            if (ub != -DBL_MAX)
+               ub -= aij->val;
+         }
+      }
+      /* encode the equality (4) */
+      n = npp_sat_encode_sum_ax(npp, row, y);
+      if (n < 0)
+         return 2; /* integer arithmetic error */
+      /* encode the condition (5) */
+      if (lb != -DBL_MAX)
+      {  rhs = (int)lb;
+         if ((double)rhs != lb)
+            return 2; /* integer arithmetic error */
+         if (npp_sat_encode_geq(npp, n, y, rhs) != 0)
+            return 1; /* original constraint is infeasible */
+      }
+      /* encode the condition (6) */
+      if (ub != +DBL_MAX)
+      {  rhs = (int)ub;
+         if ((double)rhs != ub)
+            return 2; /* integer arithmetic error */
+         if (npp_sat_encode_leq(npp, n, y, rhs) != 0)
+            return 1; /* original constraint is infeasible */
+      }
+      /* remove the specified row from the problem */
+      npp_del_row(npp, row);
+      return 0;
+}
+
+/***********************************************************************
+*  npp_sat_encode_prob - encode 0-1 feasibility problem
+*
+*  This routine translates the specified 0-1 feasibility problem to an
+*  equivalent SAT-CNF problem.
+*
+*  N.B. Currently this is a very crude implementation.
+*
+*  RETURNS
+*
+*  0           success;
+*
+*  GLP_ENOPFS  primal/integer infeasibility detected;
+*
+*  GLP_ERANGE  integer overflow occured. */
+
+int npp_sat_encode_prob(NPP *npp)
+{     NPPROW *row, *next_row, *prev_row;
+      NPPCOL *col, *next_col;
+      int cover = 0, pack = 0, partn = 0, ret;
+      /* process and remove free rows */
+      for (row = npp->r_head; row != NULL; row = next_row)
+      {  next_row = row->next;
+         if (row->lb == -DBL_MAX && row->ub == +DBL_MAX)
+            npp_sat_free_row(npp, row);
+      }
+      /* process and remove fixed columns */
+      for (col = npp->c_head; col != NULL; col = next_col)
+      {  next_col = col->next;
+         if (col->lb == col->ub)
+            xassert(npp_sat_fixed_col(npp, col) == 0);
+      }
+      /* only binary variables should remain */
+      for (col = npp->c_head; col != NULL; col = col->next)
+         xassert(col->is_int && col->lb == 0.0 && col->ub == 1.0);
+      /* new rows may be added to the end of the row list, so we walk
+         from the end to beginning of the list */
+      for (row = npp->r_tail; row != NULL; row = prev_row)
+      {  prev_row = row->prev;
+         /* process special cases */
+         ret = npp_sat_is_cover_ineq(npp, row);
+         if (ret != 0)
+         {  /* row is covering inequality */
+            cover++;
+            /* since it already encodes a clause, just transform it to
+               canonical form */
+            if (ret == 2)
+            {  xassert(npp_sat_reverse_row(npp, row) == 0);
+               ret = npp_sat_is_cover_ineq(npp, row);
+            }
+            xassert(ret == 1);
+            continue;
+         }
+         ret = npp_sat_is_partn_eq(npp, row);
+         if (ret != 0)
+         {  /* row is partitioning equality */
+            NPPROW *cov;
+            NPPAIJ *aij;
+            partn++;
+            /* transform it to canonical form */
+            if (ret == 2)
+            {  xassert(npp_sat_reverse_row(npp, row) == 0);
+               ret = npp_sat_is_partn_eq(npp, row);
+            }
+            xassert(ret == 1);
+            /* and split it into covering and packing inequalities,
+               both in canonical forms */
+            cov = npp_add_row(npp);
+            cov->lb = row->lb, cov->ub = +DBL_MAX;
+            for (aij = row->ptr; aij != NULL; aij = aij->r_next)
+               npp_add_aij(npp, cov, aij->col, aij->val);
+            xassert(npp_sat_is_cover_ineq(npp, cov) == 1);
+            /* the cover inequality already encodes a clause and do
+               not need any further processing */
+            row->lb = -DBL_MAX;
+            xassert(npp_sat_is_pack_ineq(npp, row) == 1);
+            /* the packing inequality will be processed below */
+            pack--;
+         }
+         ret = npp_sat_is_pack_ineq(npp, row);
+         if (ret != 0)
+         {  /* row is packing inequality */
+            NPPROW *rrr;
+            int nlit, desired_nlit = 4;
+            pack++;
+            /* transform it to canonical form */
+            if (ret == 2)
+            {  xassert(npp_sat_reverse_row(npp, row) == 0);
+               ret = npp_sat_is_pack_ineq(npp, row);
+            }
+            xassert(ret == 1);
+            /* process the packing inequality */
+            for (;;)
+            {  /* determine the number of literals in the remaining
+                  inequality */
+               nlit = npp_row_nnz(npp, row);
+               if (nlit <= desired_nlit)
+                  break;
+               /* split the current inequality into one having not more
+                  than desired_nlit literals and remaining one */
+               rrr = npp_sat_split_pack(npp, row, desired_nlit-1);
+               /* translate the former inequality to CNF and remove it
+                  from the original problem */
+               npp_sat_encode_pack(npp, rrr);
+            }
+            /* translate the remaining inequality to CNF and remove it
+               from the original problem */
+            npp_sat_encode_pack(npp, row);
+            continue;
+         }
+         /* translate row of general type to CNF and remove it from the
+            original problem */
+         ret = npp_sat_encode_row(npp, row);
+         if (ret == 0)
+            ;
+         else if (ret == 1)
+            ret = GLP_ENOPFS;
+         else if (ret == 2)
+            ret = GLP_ERANGE;
+         else
+            xassert(ret != ret);
+         if (ret != 0)
+            goto done;
+      }
+      ret = 0;
+      if (cover != 0)
+         xprintf("%d covering inequalities\n", cover);
+      if (pack != 0)
+         xprintf("%d packing inequalities\n", pack);
+      if (partn != 0)
+         xprintf("%d partitioning equalities\n", partn);
+done: return ret;
+}
+
+/* eof */