COIN-OR::LEMON - Graph Library

source: lemon-1.2/lemon/network_simplex.h @ 788:c92296660262

Last change on this file since 788:c92296660262 was 788:c92296660262, checked in by Alpar Juttner <alpar@…>, 10 years ago

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2009
6 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport
7 * (Egervary Research Group on Combinatorial Optimization, EGRES).
8 *
9 * Permission to use, modify and distribute this software is granted
10 * provided that this copyright notice appears in all copies. For
11 * precise terms see the accompanying LICENSE file.
12 *
13 * This software is provided "AS IS" with no warranty of any kind,
14 * express or implied, and with no claim as to its suitability for any
15 * purpose.
16 *
17 */
18
19#ifndef LEMON_NETWORK_SIMPLEX_H
20#define LEMON_NETWORK_SIMPLEX_H
21
22/// \ingroup min_cost_flow_algs
23///
24/// \file
25/// \brief Network Simplex algorithm for finding a minimum cost flow.
26
27#include <vector>
28#include <limits>
29#include <algorithm>
30
31#include <lemon/core.h>
32#include <lemon/math.h>
33
34namespace lemon {
35
36  /// \addtogroup min_cost_flow_algs
37  /// @{
38
39  /// \brief Implementation of the primal Network Simplex algorithm
40  /// for finding a \ref min_cost_flow "minimum cost flow".
41  ///
42  /// \ref NetworkSimplex implements the primal Network Simplex algorithm
43  /// for finding a \ref min_cost_flow "minimum cost flow"
44  /// \ref amo93networkflows, \ref dantzig63linearprog,
45  /// \ref kellyoneill91netsimplex.
46  /// This algorithm is a specialized version of the linear programming
47  /// simplex method directly for the minimum cost flow problem.
48  /// It is one of the most efficient solution methods.
49  ///
50  /// In general this class is the fastest implementation available
51  /// in LEMON for the minimum cost flow problem.
52  /// Moreover it supports both directions of the supply/demand inequality
53  /// constraints. For more information, see \ref SupplyType.
54  ///
55  /// Most of the parameters of the problem (except for the digraph)
56  /// can be given using separate functions, and the algorithm can be
57  /// executed using the \ref run() function. If some parameters are not
58  /// specified, then default values will be used.
59  ///
60  /// \tparam GR The digraph type the algorithm runs on.
61  /// \tparam V The value type used for flow amounts, capacity bounds
62  /// and supply values in the algorithm. By default, it is \c int.
63  /// \tparam C The value type used for costs and potentials in the
64  /// algorithm. By default, it is the same as \c V.
65  ///
66  /// \warning Both value types must be signed and all input data must
67  /// be integer.
68  ///
69  /// \note %NetworkSimplex provides five different pivot rule
70  /// implementations, from which the most efficient one is used
71  /// by default. For more information, see \ref PivotRule.
72  template <typename GR, typename V = int, typename C = V>
73  class NetworkSimplex
74  {
75  public:
76
77    /// The type of the flow amounts, capacity bounds and supply values
78    typedef V Value;
79    /// The type of the arc costs
80    typedef C Cost;
81
82  public:
83
84    /// \brief Problem type constants for the \c run() function.
85    ///
86    /// Enum type containing the problem type constants that can be
87    /// returned by the \ref run() function of the algorithm.
88    enum ProblemType {
89      /// The problem has no feasible solution (flow).
90      INFEASIBLE,
91      /// The problem has optimal solution (i.e. it is feasible and
92      /// bounded), and the algorithm has found optimal flow and node
93      /// potentials (primal and dual solutions).
94      OPTIMAL,
95      /// The objective function of the problem is unbounded, i.e.
96      /// there is a directed cycle having negative total cost and
97      /// infinite upper bound.
98      UNBOUNDED
99    };
100   
101    /// \brief Constants for selecting the type of the supply constraints.
102    ///
103    /// Enum type containing constants for selecting the supply type,
104    /// i.e. the direction of the inequalities in the supply/demand
105    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
106    ///
107    /// The default supply type is \c GEQ, the \c LEQ type can be
108    /// selected using \ref supplyType().
109    /// The equality form is a special case of both supply types.
110    enum SupplyType {
111      /// This option means that there are <em>"greater or equal"</em>
112      /// supply/demand constraints in the definition of the problem.
113      GEQ,
114      /// This option means that there are <em>"less or equal"</em>
115      /// supply/demand constraints in the definition of the problem.
116      LEQ
117    };
118   
119    /// \brief Constants for selecting the pivot rule.
120    ///
121    /// Enum type containing constants for selecting the pivot rule for
122    /// the \ref run() function.
123    ///
124    /// \ref NetworkSimplex provides five different pivot rule
125    /// implementations that significantly affect the running time
126    /// of the algorithm.
127    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
128    /// proved to be the most efficient and the most robust on various
129    /// test inputs according to our benchmark tests.
130    /// However, another pivot rule can be selected using the \ref run()
131    /// function with the proper parameter.
132    enum PivotRule {
133
134      /// The \e First \e Eligible pivot rule.
135      /// The next eligible arc is selected in a wraparound fashion
136      /// in every iteration.
137      FIRST_ELIGIBLE,
138
139      /// The \e Best \e Eligible pivot rule.
140      /// The best eligible arc is selected in every iteration.
141      BEST_ELIGIBLE,
142
143      /// The \e Block \e Search pivot rule.
144      /// A specified number of arcs are examined in every iteration
145      /// in a wraparound fashion and the best eligible arc is selected
146      /// from this block.
147      BLOCK_SEARCH,
148
149      /// The \e Candidate \e List pivot rule.
150      /// In a major iteration a candidate list is built from eligible arcs
151      /// in a wraparound fashion and in the following minor iterations
152      /// the best eligible arc is selected from this list.
153      CANDIDATE_LIST,
154
155      /// The \e Altering \e Candidate \e List pivot rule.
156      /// It is a modified version of the Candidate List method.
157      /// It keeps only the several best eligible arcs from the former
158      /// candidate list and extends this list in every iteration.
159      ALTERING_LIST
160    };
161   
162  private:
163
164    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
165
166    typedef std::vector<int> IntVector;
167    typedef std::vector<bool> BoolVector;
168    typedef std::vector<Value> ValueVector;
169    typedef std::vector<Cost> CostVector;
170
171    // State constants for arcs
172    enum ArcStateEnum {
173      STATE_UPPER = -1,
174      STATE_TREE  =  0,
175      STATE_LOWER =  1
176    };
177
178  private:
179
180    // Data related to the underlying digraph
181    const GR &_graph;
182    int _node_num;
183    int _arc_num;
184    int _all_arc_num;
185    int _search_arc_num;
186
187    // Parameters of the problem
188    bool _have_lower;
189    SupplyType _stype;
190    Value _sum_supply;
191
192    // Data structures for storing the digraph
193    IntNodeMap _node_id;
194    IntArcMap _arc_id;
195    IntVector _source;
196    IntVector _target;
197
198    // Node and arc data
199    ValueVector _lower;
200    ValueVector _upper;
201    ValueVector _cap;
202    CostVector _cost;
203    ValueVector _supply;
204    ValueVector _flow;
205    CostVector _pi;
206
207    // Data for storing the spanning tree structure
208    IntVector _parent;
209    IntVector _pred;
210    IntVector _thread;
211    IntVector _rev_thread;
212    IntVector _succ_num;
213    IntVector _last_succ;
214    IntVector _dirty_revs;
215    BoolVector _forward;
216    IntVector _state;
217    int _root;
218
219    // Temporary data used in the current pivot iteration
220    int in_arc, join, u_in, v_in, u_out, v_out;
221    int first, second, right, last;
222    int stem, par_stem, new_stem;
223    Value delta;
224
225  public:
226 
227    /// \brief Constant for infinite upper bounds (capacities).
228    ///
229    /// Constant for infinite upper bounds (capacities).
230    /// It is \c std::numeric_limits<Value>::infinity() if available,
231    /// \c std::numeric_limits<Value>::max() otherwise.
232    const Value INF;
233
234  private:
235
236    // Implementation of the First Eligible pivot rule
237    class FirstEligiblePivotRule
238    {
239    private:
240
241      // References to the NetworkSimplex class
242      const IntVector  &_source;
243      const IntVector  &_target;
244      const CostVector &_cost;
245      const IntVector  &_state;
246      const CostVector &_pi;
247      int &_in_arc;
248      int _search_arc_num;
249
250      // Pivot rule data
251      int _next_arc;
252
253    public:
254
255      // Constructor
256      FirstEligiblePivotRule(NetworkSimplex &ns) :
257        _source(ns._source), _target(ns._target),
258        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
259        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
260        _next_arc(0)
261      {}
262
263      // Find next entering arc
264      bool findEnteringArc() {
265        Cost c;
266        for (int e = _next_arc; e < _search_arc_num; ++e) {
267          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
268          if (c < 0) {
269            _in_arc = e;
270            _next_arc = e + 1;
271            return true;
272          }
273        }
274        for (int e = 0; e < _next_arc; ++e) {
275          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
276          if (c < 0) {
277            _in_arc = e;
278            _next_arc = e + 1;
279            return true;
280          }
281        }
282        return false;
283      }
284
285    }; //class FirstEligiblePivotRule
286
287
288    // Implementation of the Best Eligible pivot rule
289    class BestEligiblePivotRule
290    {
291    private:
292
293      // References to the NetworkSimplex class
294      const IntVector  &_source;
295      const IntVector  &_target;
296      const CostVector &_cost;
297      const IntVector  &_state;
298      const CostVector &_pi;
299      int &_in_arc;
300      int _search_arc_num;
301
302    public:
303
304      // Constructor
305      BestEligiblePivotRule(NetworkSimplex &ns) :
306        _source(ns._source), _target(ns._target),
307        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
308        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
309      {}
310
311      // Find next entering arc
312      bool findEnteringArc() {
313        Cost c, min = 0;
314        for (int e = 0; e < _search_arc_num; ++e) {
315          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
316          if (c < min) {
317            min = c;
318            _in_arc = e;
319          }
320        }
321        return min < 0;
322      }
323
324    }; //class BestEligiblePivotRule
325
326
327    // Implementation of the Block Search pivot rule
328    class BlockSearchPivotRule
329    {
330    private:
331
332      // References to the NetworkSimplex class
333      const IntVector  &_source;
334      const IntVector  &_target;
335      const CostVector &_cost;
336      const IntVector  &_state;
337      const CostVector &_pi;
338      int &_in_arc;
339      int _search_arc_num;
340
341      // Pivot rule data
342      int _block_size;
343      int _next_arc;
344
345    public:
346
347      // Constructor
348      BlockSearchPivotRule(NetworkSimplex &ns) :
349        _source(ns._source), _target(ns._target),
350        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
351        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
352        _next_arc(0)
353      {
354        // The main parameters of the pivot rule
355        const double BLOCK_SIZE_FACTOR = 0.5;
356        const int MIN_BLOCK_SIZE = 10;
357
358        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
359                                    std::sqrt(double(_search_arc_num))),
360                                MIN_BLOCK_SIZE );
361      }
362
363      // Find next entering arc
364      bool findEnteringArc() {
365        Cost c, min = 0;
366        int cnt = _block_size;
367        int e;
368        for (e = _next_arc; e < _search_arc_num; ++e) {
369          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
370          if (c < min) {
371            min = c;
372            _in_arc = e;
373          }
374          if (--cnt == 0) {
375            if (min < 0) goto search_end;
376            cnt = _block_size;
377          }
378        }
379        for (e = 0; e < _next_arc; ++e) {
380          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
381          if (c < min) {
382            min = c;
383            _in_arc = e;
384          }
385          if (--cnt == 0) {
386            if (min < 0) goto search_end;
387            cnt = _block_size;
388          }
389        }
390        if (min >= 0) return false;
391
392      search_end:
393        _next_arc = e;
394        return true;
395      }
396
397    }; //class BlockSearchPivotRule
398
399
400    // Implementation of the Candidate List pivot rule
401    class CandidateListPivotRule
402    {
403    private:
404
405      // References to the NetworkSimplex class
406      const IntVector  &_source;
407      const IntVector  &_target;
408      const CostVector &_cost;
409      const IntVector  &_state;
410      const CostVector &_pi;
411      int &_in_arc;
412      int _search_arc_num;
413
414      // Pivot rule data
415      IntVector _candidates;
416      int _list_length, _minor_limit;
417      int _curr_length, _minor_count;
418      int _next_arc;
419
420    public:
421
422      /// Constructor
423      CandidateListPivotRule(NetworkSimplex &ns) :
424        _source(ns._source), _target(ns._target),
425        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
426        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
427        _next_arc(0)
428      {
429        // The main parameters of the pivot rule
430        const double LIST_LENGTH_FACTOR = 0.25;
431        const int MIN_LIST_LENGTH = 10;
432        const double MINOR_LIMIT_FACTOR = 0.1;
433        const int MIN_MINOR_LIMIT = 3;
434
435        _list_length = std::max( int(LIST_LENGTH_FACTOR *
436                                     std::sqrt(double(_search_arc_num))),
437                                 MIN_LIST_LENGTH );
438        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
439                                 MIN_MINOR_LIMIT );
440        _curr_length = _minor_count = 0;
441        _candidates.resize(_list_length);
442      }
443
444      /// Find next entering arc
445      bool findEnteringArc() {
446        Cost min, c;
447        int e;
448        if (_curr_length > 0 && _minor_count < _minor_limit) {
449          // Minor iteration: select the best eligible arc from the
450          // current candidate list
451          ++_minor_count;
452          min = 0;
453          for (int i = 0; i < _curr_length; ++i) {
454            e = _candidates[i];
455            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
456            if (c < min) {
457              min = c;
458              _in_arc = e;
459            }
460            else if (c >= 0) {
461              _candidates[i--] = _candidates[--_curr_length];
462            }
463          }
464          if (min < 0) return true;
465        }
466
467        // Major iteration: build a new candidate list
468        min = 0;
469        _curr_length = 0;
470        for (e = _next_arc; e < _search_arc_num; ++e) {
471          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
472          if (c < 0) {
473            _candidates[_curr_length++] = e;
474            if (c < min) {
475              min = c;
476              _in_arc = e;
477            }
478            if (_curr_length == _list_length) goto search_end;
479          }
480        }
481        for (e = 0; e < _next_arc; ++e) {
482          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
483          if (c < 0) {
484            _candidates[_curr_length++] = e;
485            if (c < min) {
486              min = c;
487              _in_arc = e;
488            }
489            if (_curr_length == _list_length) goto search_end;
490          }
491        }
492        if (_curr_length == 0) return false;
493     
494      search_end:       
495        _minor_count = 1;
496        _next_arc = e;
497        return true;
498      }
499
500    }; //class CandidateListPivotRule
501
502
503    // Implementation of the Altering Candidate List pivot rule
504    class AlteringListPivotRule
505    {
506    private:
507
508      // References to the NetworkSimplex class
509      const IntVector  &_source;
510      const IntVector  &_target;
511      const CostVector &_cost;
512      const IntVector  &_state;
513      const CostVector &_pi;
514      int &_in_arc;
515      int _search_arc_num;
516
517      // Pivot rule data
518      int _block_size, _head_length, _curr_length;
519      int _next_arc;
520      IntVector _candidates;
521      CostVector _cand_cost;
522
523      // Functor class to compare arcs during sort of the candidate list
524      class SortFunc
525      {
526      private:
527        const CostVector &_map;
528      public:
529        SortFunc(const CostVector &map) : _map(map) {}
530        bool operator()(int left, int right) {
531          return _map[left] > _map[right];
532        }
533      };
534
535      SortFunc _sort_func;
536
537    public:
538
539      // Constructor
540      AlteringListPivotRule(NetworkSimplex &ns) :
541        _source(ns._source), _target(ns._target),
542        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
543        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
544        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
545      {
546        // The main parameters of the pivot rule
547        const double BLOCK_SIZE_FACTOR = 1.0;
548        const int MIN_BLOCK_SIZE = 10;
549        const double HEAD_LENGTH_FACTOR = 0.1;
550        const int MIN_HEAD_LENGTH = 3;
551
552        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
553                                    std::sqrt(double(_search_arc_num))),
554                                MIN_BLOCK_SIZE );
555        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
556                                 MIN_HEAD_LENGTH );
557        _candidates.resize(_head_length + _block_size);
558        _curr_length = 0;
559      }
560
561      // Find next entering arc
562      bool findEnteringArc() {
563        // Check the current candidate list
564        int e;
565        for (int i = 0; i < _curr_length; ++i) {
566          e = _candidates[i];
567          _cand_cost[e] = _state[e] *
568            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
569          if (_cand_cost[e] >= 0) {
570            _candidates[i--] = _candidates[--_curr_length];
571          }
572        }
573
574        // Extend the list
575        int cnt = _block_size;
576        int limit = _head_length;
577
578        for (e = _next_arc; e < _search_arc_num; ++e) {
579          _cand_cost[e] = _state[e] *
580            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
581          if (_cand_cost[e] < 0) {
582            _candidates[_curr_length++] = e;
583          }
584          if (--cnt == 0) {
585            if (_curr_length > limit) goto search_end;
586            limit = 0;
587            cnt = _block_size;
588          }
589        }
590        for (e = 0; e < _next_arc; ++e) {
591          _cand_cost[e] = _state[e] *
592            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
593          if (_cand_cost[e] < 0) {
594            _candidates[_curr_length++] = e;
595          }
596          if (--cnt == 0) {
597            if (_curr_length > limit) goto search_end;
598            limit = 0;
599            cnt = _block_size;
600          }
601        }
602        if (_curr_length == 0) return false;
603       
604      search_end:
605
606        // Make heap of the candidate list (approximating a partial sort)
607        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
608                   _sort_func );
609
610        // Pop the first element of the heap
611        _in_arc = _candidates[0];
612        _next_arc = e;
613        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
614                  _sort_func );
615        _curr_length = std::min(_head_length, _curr_length - 1);
616        return true;
617      }
618
619    }; //class AlteringListPivotRule
620
621  public:
622
623    /// \brief Constructor.
624    ///
625    /// The constructor of the class.
626    ///
627    /// \param graph The digraph the algorithm runs on.
628    /// \param arc_mixing Indicate if the arcs have to be stored in a
629    /// mixed order in the internal data structure.
630    /// In special cases, it could lead to better overall performance,
631    /// but it is usually slower. Therefore it is disabled by default.
632    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
633      _graph(graph), _node_id(graph), _arc_id(graph),
634      INF(std::numeric_limits<Value>::has_infinity ?
635          std::numeric_limits<Value>::infinity() :
636          std::numeric_limits<Value>::max())
637    {
638      // Check the value types
639      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
640        "The flow type of NetworkSimplex must be signed");
641      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
642        "The cost type of NetworkSimplex must be signed");
643       
644      // Resize vectors
645      _node_num = countNodes(_graph);
646      _arc_num = countArcs(_graph);
647      int all_node_num = _node_num + 1;
648      int max_arc_num = _arc_num + 2 * _node_num;
649
650      _source.resize(max_arc_num);
651      _target.resize(max_arc_num);
652
653      _lower.resize(_arc_num);
654      _upper.resize(_arc_num);
655      _cap.resize(max_arc_num);
656      _cost.resize(max_arc_num);
657      _supply.resize(all_node_num);
658      _flow.resize(max_arc_num);
659      _pi.resize(all_node_num);
660
661      _parent.resize(all_node_num);
662      _pred.resize(all_node_num);
663      _forward.resize(all_node_num);
664      _thread.resize(all_node_num);
665      _rev_thread.resize(all_node_num);
666      _succ_num.resize(all_node_num);
667      _last_succ.resize(all_node_num);
668      _state.resize(max_arc_num);
669
670      // Copy the graph
671      int i = 0;
672      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
673        _node_id[n] = i;
674      }
675      if (arc_mixing) {
676        // Store the arcs in a mixed order
677        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
678        int i = 0, j = 0;
679        for (ArcIt a(_graph); a != INVALID; ++a) {
680          _arc_id[a] = i;
681          _source[i] = _node_id[_graph.source(a)];
682          _target[i] = _node_id[_graph.target(a)];
683          if ((i += k) >= _arc_num) i = ++j;
684        }
685      } else {
686        // Store the arcs in the original order
687        int i = 0;
688        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
689          _arc_id[a] = i;
690          _source[i] = _node_id[_graph.source(a)];
691          _target[i] = _node_id[_graph.target(a)];
692        }
693      }
694     
695      // Reset parameters
696      reset();
697    }
698
699    /// \name Parameters
700    /// The parameters of the algorithm can be specified using these
701    /// functions.
702
703    /// @{
704
705    /// \brief Set the lower bounds on the arcs.
706    ///
707    /// This function sets the lower bounds on the arcs.
708    /// If it is not used before calling \ref run(), the lower bounds
709    /// will be set to zero on all arcs.
710    ///
711    /// \param map An arc map storing the lower bounds.
712    /// Its \c Value type must be convertible to the \c Value type
713    /// of the algorithm.
714    ///
715    /// \return <tt>(*this)</tt>
716    template <typename LowerMap>
717    NetworkSimplex& lowerMap(const LowerMap& map) {
718      _have_lower = true;
719      for (ArcIt a(_graph); a != INVALID; ++a) {
720        _lower[_arc_id[a]] = map[a];
721      }
722      return *this;
723    }
724
725    /// \brief Set the upper bounds (capacities) on the arcs.
726    ///
727    /// This function sets the upper bounds (capacities) on the arcs.
728    /// If it is not used before calling \ref run(), the upper bounds
729    /// will be set to \ref INF on all arcs (i.e. the flow value will be
730    /// unbounded from above on each arc).
731    ///
732    /// \param map An arc map storing the upper bounds.
733    /// Its \c Value type must be convertible to the \c Value type
734    /// of the algorithm.
735    ///
736    /// \return <tt>(*this)</tt>
737    template<typename UpperMap>
738    NetworkSimplex& upperMap(const UpperMap& map) {
739      for (ArcIt a(_graph); a != INVALID; ++a) {
740        _upper[_arc_id[a]] = map[a];
741      }
742      return *this;
743    }
744
745    /// \brief Set the costs of the arcs.
746    ///
747    /// This function sets the costs of the arcs.
748    /// If it is not used before calling \ref run(), the costs
749    /// will be set to \c 1 on all arcs.
750    ///
751    /// \param map An arc map storing the costs.
752    /// Its \c Value type must be convertible to the \c Cost type
753    /// of the algorithm.
754    ///
755    /// \return <tt>(*this)</tt>
756    template<typename CostMap>
757    NetworkSimplex& costMap(const CostMap& map) {
758      for (ArcIt a(_graph); a != INVALID; ++a) {
759        _cost[_arc_id[a]] = map[a];
760      }
761      return *this;
762    }
763
764    /// \brief Set the supply values of the nodes.
765    ///
766    /// This function sets the supply values of the nodes.
767    /// If neither this function nor \ref stSupply() is used before
768    /// calling \ref run(), the supply of each node will be set to zero.
769    ///
770    /// \param map A node map storing the supply values.
771    /// Its \c Value type must be convertible to the \c Value type
772    /// of the algorithm.
773    ///
774    /// \return <tt>(*this)</tt>
775    template<typename SupplyMap>
776    NetworkSimplex& supplyMap(const SupplyMap& map) {
777      for (NodeIt n(_graph); n != INVALID; ++n) {
778        _supply[_node_id[n]] = map[n];
779      }
780      return *this;
781    }
782
783    /// \brief Set single source and target nodes and a supply value.
784    ///
785    /// This function sets a single source node and a single target node
786    /// and the required flow value.
787    /// If neither this function nor \ref supplyMap() is used before
788    /// calling \ref run(), the supply of each node will be set to zero.
789    ///
790    /// Using this function has the same effect as using \ref supplyMap()
791    /// with such a map in which \c k is assigned to \c s, \c -k is
792    /// assigned to \c t and all other nodes have zero supply value.
793    ///
794    /// \param s The source node.
795    /// \param t The target node.
796    /// \param k The required amount of flow from node \c s to node \c t
797    /// (i.e. the supply of \c s and the demand of \c t).
798    ///
799    /// \return <tt>(*this)</tt>
800    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
801      for (int i = 0; i != _node_num; ++i) {
802        _supply[i] = 0;
803      }
804      _supply[_node_id[s]] =  k;
805      _supply[_node_id[t]] = -k;
806      return *this;
807    }
808   
809    /// \brief Set the type of the supply constraints.
810    ///
811    /// This function sets the type of the supply/demand constraints.
812    /// If it is not used before calling \ref run(), the \ref GEQ supply
813    /// type will be used.
814    ///
815    /// For more information, see \ref SupplyType.
816    ///
817    /// \return <tt>(*this)</tt>
818    NetworkSimplex& supplyType(SupplyType supply_type) {
819      _stype = supply_type;
820      return *this;
821    }
822
823    /// @}
824
825    /// \name Execution Control
826    /// The algorithm can be executed using \ref run().
827
828    /// @{
829
830    /// \brief Run the algorithm.
831    ///
832    /// This function runs the algorithm.
833    /// The paramters can be specified using functions \ref lowerMap(),
834    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
835    /// \ref supplyType().
836    /// For example,
837    /// \code
838    ///   NetworkSimplex<ListDigraph> ns(graph);
839    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
840    ///     .supplyMap(sup).run();
841    /// \endcode
842    ///
843    /// This function can be called more than once. All the parameters
844    /// that have been given are kept for the next call, unless
845    /// \ref reset() is called, thus only the modified parameters
846    /// have to be set again. See \ref reset() for examples.
847    /// However, the underlying digraph must not be modified after this
848    /// class have been constructed, since it copies and extends the graph.
849    ///
850    /// \param pivot_rule The pivot rule that will be used during the
851    /// algorithm. For more information, see \ref PivotRule.
852    ///
853    /// \return \c INFEASIBLE if no feasible flow exists,
854    /// \n \c OPTIMAL if the problem has optimal solution
855    /// (i.e. it is feasible and bounded), and the algorithm has found
856    /// optimal flow and node potentials (primal and dual solutions),
857    /// \n \c UNBOUNDED if the objective function of the problem is
858    /// unbounded, i.e. there is a directed cycle having negative total
859    /// cost and infinite upper bound.
860    ///
861    /// \see ProblemType, PivotRule
862    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
863      if (!init()) return INFEASIBLE;
864      return start(pivot_rule);
865    }
866
867    /// \brief Reset all the parameters that have been given before.
868    ///
869    /// This function resets all the paramaters that have been given
870    /// before using functions \ref lowerMap(), \ref upperMap(),
871    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
872    ///
873    /// It is useful for multiple run() calls. If this function is not
874    /// used, all the parameters given before are kept for the next
875    /// \ref run() call.
876    /// However, the underlying digraph must not be modified after this
877    /// class have been constructed, since it copies and extends the graph.
878    ///
879    /// For example,
880    /// \code
881    ///   NetworkSimplex<ListDigraph> ns(graph);
882    ///
883    ///   // First run
884    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
885    ///     .supplyMap(sup).run();
886    ///
887    ///   // Run again with modified cost map (reset() is not called,
888    ///   // so only the cost map have to be set again)
889    ///   cost[e] += 100;
890    ///   ns.costMap(cost).run();
891    ///
892    ///   // Run again from scratch using reset()
893    ///   // (the lower bounds will be set to zero on all arcs)
894    ///   ns.reset();
895    ///   ns.upperMap(capacity).costMap(cost)
896    ///     .supplyMap(sup).run();
897    /// \endcode
898    ///
899    /// \return <tt>(*this)</tt>
900    NetworkSimplex& reset() {
901      for (int i = 0; i != _node_num; ++i) {
902        _supply[i] = 0;
903      }
904      for (int i = 0; i != _arc_num; ++i) {
905        _lower[i] = 0;
906        _upper[i] = INF;
907        _cost[i] = 1;
908      }
909      _have_lower = false;
910      _stype = GEQ;
911      return *this;
912    }
913
914    /// @}
915
916    /// \name Query Functions
917    /// The results of the algorithm can be obtained using these
918    /// functions.\n
919    /// The \ref run() function must be called before using them.
920
921    /// @{
922
923    /// \brief Return the total cost of the found flow.
924    ///
925    /// This function returns the total cost of the found flow.
926    /// Its complexity is O(e).
927    ///
928    /// \note The return type of the function can be specified as a
929    /// template parameter. For example,
930    /// \code
931    ///   ns.totalCost<double>();
932    /// \endcode
933    /// It is useful if the total cost cannot be stored in the \c Cost
934    /// type of the algorithm, which is the default return type of the
935    /// function.
936    ///
937    /// \pre \ref run() must be called before using this function.
938    template <typename Number>
939    Number totalCost() const {
940      Number c = 0;
941      for (ArcIt a(_graph); a != INVALID; ++a) {
942        int i = _arc_id[a];
943        c += Number(_flow[i]) * Number(_cost[i]);
944      }
945      return c;
946    }
947
948#ifndef DOXYGEN
949    Cost totalCost() const {
950      return totalCost<Cost>();
951    }
952#endif
953
954    /// \brief Return the flow on the given arc.
955    ///
956    /// This function returns the flow on the given arc.
957    ///
958    /// \pre \ref run() must be called before using this function.
959    Value flow(const Arc& a) const {
960      return _flow[_arc_id[a]];
961    }
962
963    /// \brief Return the flow map (the primal solution).
964    ///
965    /// This function copies the flow value on each arc into the given
966    /// map. The \c Value type of the algorithm must be convertible to
967    /// the \c Value type of the map.
968    ///
969    /// \pre \ref run() must be called before using this function.
970    template <typename FlowMap>
971    void flowMap(FlowMap &map) const {
972      for (ArcIt a(_graph); a != INVALID; ++a) {
973        map.set(a, _flow[_arc_id[a]]);
974      }
975    }
976
977    /// \brief Return the potential (dual value) of the given node.
978    ///
979    /// This function returns the potential (dual value) of the
980    /// given node.
981    ///
982    /// \pre \ref run() must be called before using this function.
983    Cost potential(const Node& n) const {
984      return _pi[_node_id[n]];
985    }
986
987    /// \brief Return the potential map (the dual solution).
988    ///
989    /// This function copies the potential (dual value) of each node
990    /// into the given map.
991    /// The \c Cost type of the algorithm must be convertible to the
992    /// \c Value type of the map.
993    ///
994    /// \pre \ref run() must be called before using this function.
995    template <typename PotentialMap>
996    void potentialMap(PotentialMap &map) const {
997      for (NodeIt n(_graph); n != INVALID; ++n) {
998        map.set(n, _pi[_node_id[n]]);
999      }
1000    }
1001
1002    /// @}
1003
1004  private:
1005
1006    // Initialize internal data structures
1007    bool init() {
1008      if (_node_num == 0) return false;
1009
1010      // Check the sum of supply values
1011      _sum_supply = 0;
1012      for (int i = 0; i != _node_num; ++i) {
1013        _sum_supply += _supply[i];
1014      }
1015      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1016             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1017
1018      // Remove non-zero lower bounds
1019      if (_have_lower) {
1020        for (int i = 0; i != _arc_num; ++i) {
1021          Value c = _lower[i];
1022          if (c >= 0) {
1023            _cap[i] = _upper[i] < INF ? _upper[i] - c : INF;
1024          } else {
1025            _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF;
1026          }
1027          _supply[_source[i]] -= c;
1028          _supply[_target[i]] += c;
1029        }
1030      } else {
1031        for (int i = 0; i != _arc_num; ++i) {
1032          _cap[i] = _upper[i];
1033        }
1034      }
1035
1036      // Initialize artifical cost
1037      Cost ART_COST;
1038      if (std::numeric_limits<Cost>::is_exact) {
1039        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1040      } else {
1041        ART_COST = std::numeric_limits<Cost>::min();
1042        for (int i = 0; i != _arc_num; ++i) {
1043          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1044        }
1045        ART_COST = (ART_COST + 1) * _node_num;
1046      }
1047
1048      // Initialize arc maps
1049      for (int i = 0; i != _arc_num; ++i) {
1050        _flow[i] = 0;
1051        _state[i] = STATE_LOWER;
1052      }
1053     
1054      // Set data for the artificial root node
1055      _root = _node_num;
1056      _parent[_root] = -1;
1057      _pred[_root] = -1;
1058      _thread[_root] = 0;
1059      _rev_thread[0] = _root;
1060      _succ_num[_root] = _node_num + 1;
1061      _last_succ[_root] = _root - 1;
1062      _supply[_root] = -_sum_supply;
1063      _pi[_root] = 0;
1064
1065      // Add artificial arcs and initialize the spanning tree data structure
1066      if (_sum_supply == 0) {
1067        // EQ supply constraints
1068        _search_arc_num = _arc_num;
1069        _all_arc_num = _arc_num + _node_num;
1070        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1071          _parent[u] = _root;
1072          _pred[u] = e;
1073          _thread[u] = u + 1;
1074          _rev_thread[u + 1] = u;
1075          _succ_num[u] = 1;
1076          _last_succ[u] = u;
1077          _cap[e] = INF;
1078          _state[e] = STATE_TREE;
1079          if (_supply[u] >= 0) {
1080            _forward[u] = true;
1081            _pi[u] = 0;
1082            _source[e] = u;
1083            _target[e] = _root;
1084            _flow[e] = _supply[u];
1085            _cost[e] = 0;
1086          } else {
1087            _forward[u] = false;
1088            _pi[u] = ART_COST;
1089            _source[e] = _root;
1090            _target[e] = u;
1091            _flow[e] = -_supply[u];
1092            _cost[e] = ART_COST;
1093          }
1094        }
1095      }
1096      else if (_sum_supply > 0) {
1097        // LEQ supply constraints
1098        _search_arc_num = _arc_num + _node_num;
1099        int f = _arc_num + _node_num;
1100        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1101          _parent[u] = _root;
1102          _thread[u] = u + 1;
1103          _rev_thread[u + 1] = u;
1104          _succ_num[u] = 1;
1105          _last_succ[u] = u;
1106          if (_supply[u] >= 0) {
1107            _forward[u] = true;
1108            _pi[u] = 0;
1109            _pred[u] = e;
1110            _source[e] = u;
1111            _target[e] = _root;
1112            _cap[e] = INF;
1113            _flow[e] = _supply[u];
1114            _cost[e] = 0;
1115            _state[e] = STATE_TREE;
1116          } else {
1117            _forward[u] = false;
1118            _pi[u] = ART_COST;
1119            _pred[u] = f;
1120            _source[f] = _root;
1121            _target[f] = u;
1122            _cap[f] = INF;
1123            _flow[f] = -_supply[u];
1124            _cost[f] = ART_COST;
1125            _state[f] = STATE_TREE;
1126            _source[e] = u;
1127            _target[e] = _root;
1128            _cap[e] = INF;
1129            _flow[e] = 0;
1130            _cost[e] = 0;
1131            _state[e] = STATE_LOWER;
1132            ++f;
1133          }
1134        }
1135        _all_arc_num = f;
1136      }
1137      else {
1138        // GEQ supply constraints
1139        _search_arc_num = _arc_num + _node_num;
1140        int f = _arc_num + _node_num;
1141        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1142          _parent[u] = _root;
1143          _thread[u] = u + 1;
1144          _rev_thread[u + 1] = u;
1145          _succ_num[u] = 1;
1146          _last_succ[u] = u;
1147          if (_supply[u] <= 0) {
1148            _forward[u] = false;
1149            _pi[u] = 0;
1150            _pred[u] = e;
1151            _source[e] = _root;
1152            _target[e] = u;
1153            _cap[e] = INF;
1154            _flow[e] = -_supply[u];
1155            _cost[e] = 0;
1156            _state[e] = STATE_TREE;
1157          } else {
1158            _forward[u] = true;
1159            _pi[u] = -ART_COST;
1160            _pred[u] = f;
1161            _source[f] = u;
1162            _target[f] = _root;
1163            _cap[f] = INF;
1164            _flow[f] = _supply[u];
1165            _state[f] = STATE_TREE;
1166            _cost[f] = ART_COST;
1167            _source[e] = _root;
1168            _target[e] = u;
1169            _cap[e] = INF;
1170            _flow[e] = 0;
1171            _cost[e] = 0;
1172            _state[e] = STATE_LOWER;
1173            ++f;
1174          }
1175        }
1176        _all_arc_num = f;
1177      }
1178
1179      return true;
1180    }
1181
1182    // Find the join node
1183    void findJoinNode() {
1184      int u = _source[in_arc];
1185      int v = _target[in_arc];
1186      while (u != v) {
1187        if (_succ_num[u] < _succ_num[v]) {
1188          u = _parent[u];
1189        } else {
1190          v = _parent[v];
1191        }
1192      }
1193      join = u;
1194    }
1195
1196    // Find the leaving arc of the cycle and returns true if the
1197    // leaving arc is not the same as the entering arc
1198    bool findLeavingArc() {
1199      // Initialize first and second nodes according to the direction
1200      // of the cycle
1201      if (_state[in_arc] == STATE_LOWER) {
1202        first  = _source[in_arc];
1203        second = _target[in_arc];
1204      } else {
1205        first  = _target[in_arc];
1206        second = _source[in_arc];
1207      }
1208      delta = _cap[in_arc];
1209      int result = 0;
1210      Value d;
1211      int e;
1212
1213      // Search the cycle along the path form the first node to the root
1214      for (int u = first; u != join; u = _parent[u]) {
1215        e = _pred[u];
1216        d = _forward[u] ?
1217          _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]);
1218        if (d < delta) {
1219          delta = d;
1220          u_out = u;
1221          result = 1;
1222        }
1223      }
1224      // Search the cycle along the path form the second node to the root
1225      for (int u = second; u != join; u = _parent[u]) {
1226        e = _pred[u];
1227        d = _forward[u] ?
1228          (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e];
1229        if (d <= delta) {
1230          delta = d;
1231          u_out = u;
1232          result = 2;
1233        }
1234      }
1235
1236      if (result == 1) {
1237        u_in = first;
1238        v_in = second;
1239      } else {
1240        u_in = second;
1241        v_in = first;
1242      }
1243      return result != 0;
1244    }
1245
1246    // Change _flow and _state vectors
1247    void changeFlow(bool change) {
1248      // Augment along the cycle
1249      if (delta > 0) {
1250        Value val = _state[in_arc] * delta;
1251        _flow[in_arc] += val;
1252        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1253          _flow[_pred[u]] += _forward[u] ? -val : val;
1254        }
1255        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1256          _flow[_pred[u]] += _forward[u] ? val : -val;
1257        }
1258      }
1259      // Update the state of the entering and leaving arcs
1260      if (change) {
1261        _state[in_arc] = STATE_TREE;
1262        _state[_pred[u_out]] =
1263          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1264      } else {
1265        _state[in_arc] = -_state[in_arc];
1266      }
1267    }
1268
1269    // Update the tree structure
1270    void updateTreeStructure() {
1271      int u, w;
1272      int old_rev_thread = _rev_thread[u_out];
1273      int old_succ_num = _succ_num[u_out];
1274      int old_last_succ = _last_succ[u_out];
1275      v_out = _parent[u_out];
1276
1277      u = _last_succ[u_in];  // the last successor of u_in
1278      right = _thread[u];    // the node after it
1279
1280      // Handle the case when old_rev_thread equals to v_in
1281      // (it also means that join and v_out coincide)
1282      if (old_rev_thread == v_in) {
1283        last = _thread[_last_succ[u_out]];
1284      } else {
1285        last = _thread[v_in];
1286      }
1287
1288      // Update _thread and _parent along the stem nodes (i.e. the nodes
1289      // between u_in and u_out, whose parent have to be changed)
1290      _thread[v_in] = stem = u_in;
1291      _dirty_revs.clear();
1292      _dirty_revs.push_back(v_in);
1293      par_stem = v_in;
1294      while (stem != u_out) {
1295        // Insert the next stem node into the thread list
1296        new_stem = _parent[stem];
1297        _thread[u] = new_stem;
1298        _dirty_revs.push_back(u);
1299
1300        // Remove the subtree of stem from the thread list
1301        w = _rev_thread[stem];
1302        _thread[w] = right;
1303        _rev_thread[right] = w;
1304
1305        // Change the parent node and shift stem nodes
1306        _parent[stem] = par_stem;
1307        par_stem = stem;
1308        stem = new_stem;
1309
1310        // Update u and right
1311        u = _last_succ[stem] == _last_succ[par_stem] ?
1312          _rev_thread[par_stem] : _last_succ[stem];
1313        right = _thread[u];
1314      }
1315      _parent[u_out] = par_stem;
1316      _thread[u] = last;
1317      _rev_thread[last] = u;
1318      _last_succ[u_out] = u;
1319
1320      // Remove the subtree of u_out from the thread list except for
1321      // the case when old_rev_thread equals to v_in
1322      // (it also means that join and v_out coincide)
1323      if (old_rev_thread != v_in) {
1324        _thread[old_rev_thread] = right;
1325        _rev_thread[right] = old_rev_thread;
1326      }
1327
1328      // Update _rev_thread using the new _thread values
1329      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1330        u = _dirty_revs[i];
1331        _rev_thread[_thread[u]] = u;
1332      }
1333
1334      // Update _pred, _forward, _last_succ and _succ_num for the
1335      // stem nodes from u_out to u_in
1336      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1337      u = u_out;
1338      while (u != u_in) {
1339        w = _parent[u];
1340        _pred[u] = _pred[w];
1341        _forward[u] = !_forward[w];
1342        tmp_sc += _succ_num[u] - _succ_num[w];
1343        _succ_num[u] = tmp_sc;
1344        _last_succ[w] = tmp_ls;
1345        u = w;
1346      }
1347      _pred[u_in] = in_arc;
1348      _forward[u_in] = (u_in == _source[in_arc]);
1349      _succ_num[u_in] = old_succ_num;
1350
1351      // Set limits for updating _last_succ form v_in and v_out
1352      // towards the root
1353      int up_limit_in = -1;
1354      int up_limit_out = -1;
1355      if (_last_succ[join] == v_in) {
1356        up_limit_out = join;
1357      } else {
1358        up_limit_in = join;
1359      }
1360
1361      // Update _last_succ from v_in towards the root
1362      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1363           u = _parent[u]) {
1364        _last_succ[u] = _last_succ[u_out];
1365      }
1366      // Update _last_succ from v_out towards the root
1367      if (join != old_rev_thread && v_in != old_rev_thread) {
1368        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1369             u = _parent[u]) {
1370          _last_succ[u] = old_rev_thread;
1371        }
1372      } else {
1373        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1374             u = _parent[u]) {
1375          _last_succ[u] = _last_succ[u_out];
1376        }
1377      }
1378
1379      // Update _succ_num from v_in to join
1380      for (u = v_in; u != join; u = _parent[u]) {
1381        _succ_num[u] += old_succ_num;
1382      }
1383      // Update _succ_num from v_out to join
1384      for (u = v_out; u != join; u = _parent[u]) {
1385        _succ_num[u] -= old_succ_num;
1386      }
1387    }
1388
1389    // Update potentials
1390    void updatePotential() {
1391      Cost sigma = _forward[u_in] ?
1392        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1393        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1394      // Update potentials in the subtree, which has been moved
1395      int end = _thread[_last_succ[u_in]];
1396      for (int u = u_in; u != end; u = _thread[u]) {
1397        _pi[u] += sigma;
1398      }
1399    }
1400
1401    // Execute the algorithm
1402    ProblemType start(PivotRule pivot_rule) {
1403      // Select the pivot rule implementation
1404      switch (pivot_rule) {
1405        case FIRST_ELIGIBLE:
1406          return start<FirstEligiblePivotRule>();
1407        case BEST_ELIGIBLE:
1408          return start<BestEligiblePivotRule>();
1409        case BLOCK_SEARCH:
1410          return start<BlockSearchPivotRule>();
1411        case CANDIDATE_LIST:
1412          return start<CandidateListPivotRule>();
1413        case ALTERING_LIST:
1414          return start<AlteringListPivotRule>();
1415      }
1416      return INFEASIBLE; // avoid warning
1417    }
1418
1419    template <typename PivotRuleImpl>
1420    ProblemType start() {
1421      PivotRuleImpl pivot(*this);
1422
1423      // Execute the Network Simplex algorithm
1424      while (pivot.findEnteringArc()) {
1425        findJoinNode();
1426        bool change = findLeavingArc();
1427        if (delta >= INF) return UNBOUNDED;
1428        changeFlow(change);
1429        if (change) {
1430          updateTreeStructure();
1431          updatePotential();
1432        }
1433      }
1434     
1435      // Check feasibility
1436      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1437        if (_flow[e] != 0) return INFEASIBLE;
1438      }
1439
1440      // Transform the solution and the supply map to the original form
1441      if (_have_lower) {
1442        for (int i = 0; i != _arc_num; ++i) {
1443          Value c = _lower[i];
1444          if (c != 0) {
1445            _flow[i] += c;
1446            _supply[_source[i]] += c;
1447            _supply[_target[i]] -= c;
1448          }
1449        }
1450      }
1451     
1452      // Shift potentials to meet the requirements of the GEQ/LEQ type
1453      // optimality conditions
1454      if (_sum_supply == 0) {
1455        if (_stype == GEQ) {
1456          Cost max_pot = std::numeric_limits<Cost>::min();
1457          for (int i = 0; i != _node_num; ++i) {
1458            if (_pi[i] > max_pot) max_pot = _pi[i];
1459          }
1460          if (max_pot > 0) {
1461            for (int i = 0; i != _node_num; ++i)
1462              _pi[i] -= max_pot;
1463          }
1464        } else {
1465          Cost min_pot = std::numeric_limits<Cost>::max();
1466          for (int i = 0; i != _node_num; ++i) {
1467            if (_pi[i] < min_pot) min_pot = _pi[i];
1468          }
1469          if (min_pot < 0) {
1470            for (int i = 0; i != _node_num; ++i)
1471              _pi[i] -= min_pot;
1472          }
1473        }
1474      }
1475
1476      return OPTIMAL;
1477    }
1478
1479  }; //class NetworkSimplex
1480
1481  ///@}
1482
1483} //namespace lemon
1484
1485#endif //LEMON_NETWORK_SIMPLEX_H
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