COIN-OR::LEMON - Graph Library

source: lemon/lemon/network_simplex.h @ 878:4b1b378823dc

Last change on this file since 878:4b1b378823dc was 878:4b1b378823dc, checked in by Peter Kovacs <kpeter@…>, 10 years ago

Small doc improvements + unifications in MCF classes (#180)

File size: 45.3 KB
Line 
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 highly efficient specialized version of the
47  /// linear programming simplex method directly for the minimum cost
48  /// flow problem.
49  ///
50  /// In general, %NetworkSimplex is the fastest implementation available
51  /// in LEMON for this 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 number type used for flow amounts, capacity bounds
62  /// and supply values in the algorithm. By default, it is \c int.
63  /// \tparam C The number type used for costs and potentials in the
64  /// algorithm. By default, it is the same as \c V.
65  ///
66  /// \warning Both number 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.
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<char> CharVector;
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    CharVector _forward;
216    CharVector _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    const Value MAX;
226
227  public:
228 
229    /// \brief Constant for infinite upper bounds (capacities).
230    ///
231    /// Constant for infinite upper bounds (capacities).
232    /// It is \c std::numeric_limits<Value>::infinity() if available,
233    /// \c std::numeric_limits<Value>::max() otherwise.
234    const Value INF;
235
236  private:
237
238    // Implementation of the First Eligible pivot rule
239    class FirstEligiblePivotRule
240    {
241    private:
242
243      // References to the NetworkSimplex class
244      const IntVector  &_source;
245      const IntVector  &_target;
246      const CostVector &_cost;
247      const CharVector &_state;
248      const CostVector &_pi;
249      int &_in_arc;
250      int _search_arc_num;
251
252      // Pivot rule data
253      int _next_arc;
254
255    public:
256
257      // Constructor
258      FirstEligiblePivotRule(NetworkSimplex &ns) :
259        _source(ns._source), _target(ns._target),
260        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
261        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
262        _next_arc(0)
263      {}
264
265      // Find next entering arc
266      bool findEnteringArc() {
267        Cost c;
268        for (int e = _next_arc; e < _search_arc_num; ++e) {
269          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
270          if (c < 0) {
271            _in_arc = e;
272            _next_arc = e + 1;
273            return true;
274          }
275        }
276        for (int e = 0; e < _next_arc; ++e) {
277          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
278          if (c < 0) {
279            _in_arc = e;
280            _next_arc = e + 1;
281            return true;
282          }
283        }
284        return false;
285      }
286
287    }; //class FirstEligiblePivotRule
288
289
290    // Implementation of the Best Eligible pivot rule
291    class BestEligiblePivotRule
292    {
293    private:
294
295      // References to the NetworkSimplex class
296      const IntVector  &_source;
297      const IntVector  &_target;
298      const CostVector &_cost;
299      const CharVector &_state;
300      const CostVector &_pi;
301      int &_in_arc;
302      int _search_arc_num;
303
304    public:
305
306      // Constructor
307      BestEligiblePivotRule(NetworkSimplex &ns) :
308        _source(ns._source), _target(ns._target),
309        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
310        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
311      {}
312
313      // Find next entering arc
314      bool findEnteringArc() {
315        Cost c, min = 0;
316        for (int e = 0; e < _search_arc_num; ++e) {
317          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
318          if (c < min) {
319            min = c;
320            _in_arc = e;
321          }
322        }
323        return min < 0;
324      }
325
326    }; //class BestEligiblePivotRule
327
328
329    // Implementation of the Block Search pivot rule
330    class BlockSearchPivotRule
331    {
332    private:
333
334      // References to the NetworkSimplex class
335      const IntVector  &_source;
336      const IntVector  &_target;
337      const CostVector &_cost;
338      const CharVector &_state;
339      const CostVector &_pi;
340      int &_in_arc;
341      int _search_arc_num;
342
343      // Pivot rule data
344      int _block_size;
345      int _next_arc;
346
347    public:
348
349      // Constructor
350      BlockSearchPivotRule(NetworkSimplex &ns) :
351        _source(ns._source), _target(ns._target),
352        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
353        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
354        _next_arc(0)
355      {
356        // The main parameters of the pivot rule
357        const double BLOCK_SIZE_FACTOR = 0.5;
358        const int MIN_BLOCK_SIZE = 10;
359
360        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
361                                    std::sqrt(double(_search_arc_num))),
362                                MIN_BLOCK_SIZE );
363      }
364
365      // Find next entering arc
366      bool findEnteringArc() {
367        Cost c, min = 0;
368        int cnt = _block_size;
369        int e;
370        for (e = _next_arc; e < _search_arc_num; ++e) {
371          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
372          if (c < min) {
373            min = c;
374            _in_arc = e;
375          }
376          if (--cnt == 0) {
377            if (min < 0) goto search_end;
378            cnt = _block_size;
379          }
380        }
381        for (e = 0; e < _next_arc; ++e) {
382          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
383          if (c < min) {
384            min = c;
385            _in_arc = e;
386          }
387          if (--cnt == 0) {
388            if (min < 0) goto search_end;
389            cnt = _block_size;
390          }
391        }
392        if (min >= 0) return false;
393
394      search_end:
395        _next_arc = e;
396        return true;
397      }
398
399    }; //class BlockSearchPivotRule
400
401
402    // Implementation of the Candidate List pivot rule
403    class CandidateListPivotRule
404    {
405    private:
406
407      // References to the NetworkSimplex class
408      const IntVector  &_source;
409      const IntVector  &_target;
410      const CostVector &_cost;
411      const CharVector &_state;
412      const CostVector &_pi;
413      int &_in_arc;
414      int _search_arc_num;
415
416      // Pivot rule data
417      IntVector _candidates;
418      int _list_length, _minor_limit;
419      int _curr_length, _minor_count;
420      int _next_arc;
421
422    public:
423
424      /// Constructor
425      CandidateListPivotRule(NetworkSimplex &ns) :
426        _source(ns._source), _target(ns._target),
427        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
428        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
429        _next_arc(0)
430      {
431        // The main parameters of the pivot rule
432        const double LIST_LENGTH_FACTOR = 0.25;
433        const int MIN_LIST_LENGTH = 10;
434        const double MINOR_LIMIT_FACTOR = 0.1;
435        const int MIN_MINOR_LIMIT = 3;
436
437        _list_length = std::max( int(LIST_LENGTH_FACTOR *
438                                     std::sqrt(double(_search_arc_num))),
439                                 MIN_LIST_LENGTH );
440        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
441                                 MIN_MINOR_LIMIT );
442        _curr_length = _minor_count = 0;
443        _candidates.resize(_list_length);
444      }
445
446      /// Find next entering arc
447      bool findEnteringArc() {
448        Cost min, c;
449        int e;
450        if (_curr_length > 0 && _minor_count < _minor_limit) {
451          // Minor iteration: select the best eligible arc from the
452          // current candidate list
453          ++_minor_count;
454          min = 0;
455          for (int i = 0; i < _curr_length; ++i) {
456            e = _candidates[i];
457            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
458            if (c < min) {
459              min = c;
460              _in_arc = e;
461            }
462            else if (c >= 0) {
463              _candidates[i--] = _candidates[--_curr_length];
464            }
465          }
466          if (min < 0) return true;
467        }
468
469        // Major iteration: build a new candidate list
470        min = 0;
471        _curr_length = 0;
472        for (e = _next_arc; e < _search_arc_num; ++e) {
473          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
474          if (c < 0) {
475            _candidates[_curr_length++] = e;
476            if (c < min) {
477              min = c;
478              _in_arc = e;
479            }
480            if (_curr_length == _list_length) goto search_end;
481          }
482        }
483        for (e = 0; e < _next_arc; ++e) {
484          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
485          if (c < 0) {
486            _candidates[_curr_length++] = e;
487            if (c < min) {
488              min = c;
489              _in_arc = e;
490            }
491            if (_curr_length == _list_length) goto search_end;
492          }
493        }
494        if (_curr_length == 0) return false;
495     
496      search_end:       
497        _minor_count = 1;
498        _next_arc = e;
499        return true;
500      }
501
502    }; //class CandidateListPivotRule
503
504
505    // Implementation of the Altering Candidate List pivot rule
506    class AlteringListPivotRule
507    {
508    private:
509
510      // References to the NetworkSimplex class
511      const IntVector  &_source;
512      const IntVector  &_target;
513      const CostVector &_cost;
514      const CharVector &_state;
515      const CostVector &_pi;
516      int &_in_arc;
517      int _search_arc_num;
518
519      // Pivot rule data
520      int _block_size, _head_length, _curr_length;
521      int _next_arc;
522      IntVector _candidates;
523      CostVector _cand_cost;
524
525      // Functor class to compare arcs during sort of the candidate list
526      class SortFunc
527      {
528      private:
529        const CostVector &_map;
530      public:
531        SortFunc(const CostVector &map) : _map(map) {}
532        bool operator()(int left, int right) {
533          return _map[left] > _map[right];
534        }
535      };
536
537      SortFunc _sort_func;
538
539    public:
540
541      // Constructor
542      AlteringListPivotRule(NetworkSimplex &ns) :
543        _source(ns._source), _target(ns._target),
544        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
545        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
546        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
547      {
548        // The main parameters of the pivot rule
549        const double BLOCK_SIZE_FACTOR = 1.0;
550        const int MIN_BLOCK_SIZE = 10;
551        const double HEAD_LENGTH_FACTOR = 0.1;
552        const int MIN_HEAD_LENGTH = 3;
553
554        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
555                                    std::sqrt(double(_search_arc_num))),
556                                MIN_BLOCK_SIZE );
557        _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size),
558                                 MIN_HEAD_LENGTH );
559        _candidates.resize(_head_length + _block_size);
560        _curr_length = 0;
561      }
562
563      // Find next entering arc
564      bool findEnteringArc() {
565        // Check the current candidate list
566        int e;
567        for (int i = 0; i < _curr_length; ++i) {
568          e = _candidates[i];
569          _cand_cost[e] = _state[e] *
570            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
571          if (_cand_cost[e] >= 0) {
572            _candidates[i--] = _candidates[--_curr_length];
573          }
574        }
575
576        // Extend the list
577        int cnt = _block_size;
578        int limit = _head_length;
579
580        for (e = _next_arc; e < _search_arc_num; ++e) {
581          _cand_cost[e] = _state[e] *
582            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
583          if (_cand_cost[e] < 0) {
584            _candidates[_curr_length++] = e;
585          }
586          if (--cnt == 0) {
587            if (_curr_length > limit) goto search_end;
588            limit = 0;
589            cnt = _block_size;
590          }
591        }
592        for (e = 0; e < _next_arc; ++e) {
593          _cand_cost[e] = _state[e] *
594            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
595          if (_cand_cost[e] < 0) {
596            _candidates[_curr_length++] = e;
597          }
598          if (--cnt == 0) {
599            if (_curr_length > limit) goto search_end;
600            limit = 0;
601            cnt = _block_size;
602          }
603        }
604        if (_curr_length == 0) return false;
605       
606      search_end:
607
608        // Make heap of the candidate list (approximating a partial sort)
609        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
610                   _sort_func );
611
612        // Pop the first element of the heap
613        _in_arc = _candidates[0];
614        _next_arc = e;
615        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
616                  _sort_func );
617        _curr_length = std::min(_head_length, _curr_length - 1);
618        return true;
619      }
620
621    }; //class AlteringListPivotRule
622
623  public:
624
625    /// \brief Constructor.
626    ///
627    /// The constructor of the class.
628    ///
629    /// \param graph The digraph the algorithm runs on.
630    /// \param arc_mixing Indicate if the arcs have to be stored in a
631    /// mixed order in the internal data structure.
632    /// In special cases, it could lead to better overall performance,
633    /// but it is usually slower. Therefore it is disabled by default.
634    NetworkSimplex(const GR& graph, bool arc_mixing = false) :
635      _graph(graph), _node_id(graph), _arc_id(graph),
636      MAX(std::numeric_limits<Value>::max()),
637      INF(std::numeric_limits<Value>::has_infinity ?
638          std::numeric_limits<Value>::infinity() : MAX)
639    {
640      // Check the number types
641      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
642        "The flow type of NetworkSimplex must be signed");
643      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
644        "The cost type of NetworkSimplex must be signed");
645       
646      // Resize vectors
647      _node_num = countNodes(_graph);
648      _arc_num = countArcs(_graph);
649      int all_node_num = _node_num + 1;
650      int max_arc_num = _arc_num + 2 * _node_num;
651
652      _source.resize(max_arc_num);
653      _target.resize(max_arc_num);
654
655      _lower.resize(_arc_num);
656      _upper.resize(_arc_num);
657      _cap.resize(max_arc_num);
658      _cost.resize(max_arc_num);
659      _supply.resize(all_node_num);
660      _flow.resize(max_arc_num);
661      _pi.resize(all_node_num);
662
663      _parent.resize(all_node_num);
664      _pred.resize(all_node_num);
665      _forward.resize(all_node_num);
666      _thread.resize(all_node_num);
667      _rev_thread.resize(all_node_num);
668      _succ_num.resize(all_node_num);
669      _last_succ.resize(all_node_num);
670      _state.resize(max_arc_num);
671
672      // Copy the graph
673      int i = 0;
674      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
675        _node_id[n] = i;
676      }
677      if (arc_mixing) {
678        // Store the arcs in a mixed order
679        int k = std::max(int(std::sqrt(double(_arc_num))), 10);
680        int i = 0, j = 0;
681        for (ArcIt a(_graph); a != INVALID; ++a) {
682          _arc_id[a] = i;
683          _source[i] = _node_id[_graph.source(a)];
684          _target[i] = _node_id[_graph.target(a)];
685          if ((i += k) >= _arc_num) i = ++j;
686        }
687      } else {
688        // Store the arcs in the original order
689        int i = 0;
690        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
691          _arc_id[a] = i;
692          _source[i] = _node_id[_graph.source(a)];
693          _target[i] = _node_id[_graph.target(a)];
694        }
695      }
696     
697      // Reset parameters
698      reset();
699    }
700
701    /// \name Parameters
702    /// The parameters of the algorithm can be specified using these
703    /// functions.
704
705    /// @{
706
707    /// \brief Set the lower bounds on the arcs.
708    ///
709    /// This function sets the lower bounds on the arcs.
710    /// If it is not used before calling \ref run(), the lower bounds
711    /// will be set to zero on all arcs.
712    ///
713    /// \param map An arc map storing the lower bounds.
714    /// Its \c Value type must be convertible to the \c Value type
715    /// of the algorithm.
716    ///
717    /// \return <tt>(*this)</tt>
718    template <typename LowerMap>
719    NetworkSimplex& lowerMap(const LowerMap& map) {
720      _have_lower = true;
721      for (ArcIt a(_graph); a != INVALID; ++a) {
722        _lower[_arc_id[a]] = map[a];
723      }
724      return *this;
725    }
726
727    /// \brief Set the upper bounds (capacities) on the arcs.
728    ///
729    /// This function sets the upper bounds (capacities) on the arcs.
730    /// If it is not used before calling \ref run(), the upper bounds
731    /// will be set to \ref INF on all arcs (i.e. the flow value will be
732    /// unbounded from above).
733    ///
734    /// \param map An arc map storing the upper bounds.
735    /// Its \c Value type must be convertible to the \c Value type
736    /// of the algorithm.
737    ///
738    /// \return <tt>(*this)</tt>
739    template<typename UpperMap>
740    NetworkSimplex& upperMap(const UpperMap& map) {
741      for (ArcIt a(_graph); a != INVALID; ++a) {
742        _upper[_arc_id[a]] = map[a];
743      }
744      return *this;
745    }
746
747    /// \brief Set the costs of the arcs.
748    ///
749    /// This function sets the costs of the arcs.
750    /// If it is not used before calling \ref run(), the costs
751    /// will be set to \c 1 on all arcs.
752    ///
753    /// \param map An arc map storing the costs.
754    /// Its \c Value type must be convertible to the \c Cost type
755    /// of the algorithm.
756    ///
757    /// \return <tt>(*this)</tt>
758    template<typename CostMap>
759    NetworkSimplex& costMap(const CostMap& map) {
760      for (ArcIt a(_graph); a != INVALID; ++a) {
761        _cost[_arc_id[a]] = map[a];
762      }
763      return *this;
764    }
765
766    /// \brief Set the supply values of the nodes.
767    ///
768    /// This function sets the supply values of the nodes.
769    /// If neither this function nor \ref stSupply() is used before
770    /// calling \ref run(), the supply of each node will be set to zero.
771    ///
772    /// \param map A node map storing the supply values.
773    /// Its \c Value type must be convertible to the \c Value type
774    /// of the algorithm.
775    ///
776    /// \return <tt>(*this)</tt>
777    template<typename SupplyMap>
778    NetworkSimplex& supplyMap(const SupplyMap& map) {
779      for (NodeIt n(_graph); n != INVALID; ++n) {
780        _supply[_node_id[n]] = map[n];
781      }
782      return *this;
783    }
784
785    /// \brief Set single source and target nodes and a supply value.
786    ///
787    /// This function sets a single source node and a single target node
788    /// and the required flow value.
789    /// If neither this function nor \ref supplyMap() is used before
790    /// calling \ref run(), the supply of each node will be set to zero.
791    ///
792    /// Using this function has the same effect as using \ref supplyMap()
793    /// with such a map in which \c k is assigned to \c s, \c -k is
794    /// assigned to \c t and all other nodes have zero supply value.
795    ///
796    /// \param s The source node.
797    /// \param t The target node.
798    /// \param k The required amount of flow from node \c s to node \c t
799    /// (i.e. the supply of \c s and the demand of \c t).
800    ///
801    /// \return <tt>(*this)</tt>
802    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
803      for (int i = 0; i != _node_num; ++i) {
804        _supply[i] = 0;
805      }
806      _supply[_node_id[s]] =  k;
807      _supply[_node_id[t]] = -k;
808      return *this;
809    }
810   
811    /// \brief Set the type of the supply constraints.
812    ///
813    /// This function sets the type of the supply/demand constraints.
814    /// If it is not used before calling \ref run(), the \ref GEQ supply
815    /// type will be used.
816    ///
817    /// For more information, see \ref SupplyType.
818    ///
819    /// \return <tt>(*this)</tt>
820    NetworkSimplex& supplyType(SupplyType supply_type) {
821      _stype = supply_type;
822      return *this;
823    }
824
825    /// @}
826
827    /// \name Execution Control
828    /// The algorithm can be executed using \ref run().
829
830    /// @{
831
832    /// \brief Run the algorithm.
833    ///
834    /// This function runs the algorithm.
835    /// The paramters can be specified using functions \ref lowerMap(),
836    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
837    /// \ref supplyType().
838    /// For example,
839    /// \code
840    ///   NetworkSimplex<ListDigraph> ns(graph);
841    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
842    ///     .supplyMap(sup).run();
843    /// \endcode
844    ///
845    /// This function can be called more than once. All the parameters
846    /// that have been given are kept for the next call, unless
847    /// \ref reset() is called, thus only the modified parameters
848    /// have to be set again. See \ref reset() for examples.
849    /// However, the underlying digraph must not be modified after this
850    /// class have been constructed, since it copies and extends the graph.
851    ///
852    /// \param pivot_rule The pivot rule that will be used during the
853    /// algorithm. For more information, see \ref PivotRule.
854    ///
855    /// \return \c INFEASIBLE if no feasible flow exists,
856    /// \n \c OPTIMAL if the problem has optimal solution
857    /// (i.e. it is feasible and bounded), and the algorithm has found
858    /// optimal flow and node potentials (primal and dual solutions),
859    /// \n \c UNBOUNDED if the objective function of the problem is
860    /// unbounded, i.e. there is a directed cycle having negative total
861    /// cost and infinite upper bound.
862    ///
863    /// \see ProblemType, PivotRule
864    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
865      if (!init()) return INFEASIBLE;
866      return start(pivot_rule);
867    }
868
869    /// \brief Reset all the parameters that have been given before.
870    ///
871    /// This function resets all the paramaters that have been given
872    /// before using functions \ref lowerMap(), \ref upperMap(),
873    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
874    ///
875    /// It is useful for multiple run() calls. If this function is not
876    /// used, all the parameters given before are kept for the next
877    /// \ref run() call.
878    /// However, the underlying digraph must not be modified after this
879    /// class have been constructed, since it copies and extends the graph.
880    ///
881    /// For example,
882    /// \code
883    ///   NetworkSimplex<ListDigraph> ns(graph);
884    ///
885    ///   // First run
886    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
887    ///     .supplyMap(sup).run();
888    ///
889    ///   // Run again with modified cost map (reset() is not called,
890    ///   // so only the cost map have to be set again)
891    ///   cost[e] += 100;
892    ///   ns.costMap(cost).run();
893    ///
894    ///   // Run again from scratch using reset()
895    ///   // (the lower bounds will be set to zero on all arcs)
896    ///   ns.reset();
897    ///   ns.upperMap(capacity).costMap(cost)
898    ///     .supplyMap(sup).run();
899    /// \endcode
900    ///
901    /// \return <tt>(*this)</tt>
902    NetworkSimplex& reset() {
903      for (int i = 0; i != _node_num; ++i) {
904        _supply[i] = 0;
905      }
906      for (int i = 0; i != _arc_num; ++i) {
907        _lower[i] = 0;
908        _upper[i] = INF;
909        _cost[i] = 1;
910      }
911      _have_lower = false;
912      _stype = GEQ;
913      return *this;
914    }
915
916    /// @}
917
918    /// \name Query Functions
919    /// The results of the algorithm can be obtained using these
920    /// functions.\n
921    /// The \ref run() function must be called before using them.
922
923    /// @{
924
925    /// \brief Return the total cost of the found flow.
926    ///
927    /// This function returns the total cost of the found flow.
928    /// Its complexity is O(e).
929    ///
930    /// \note The return type of the function can be specified as a
931    /// template parameter. For example,
932    /// \code
933    ///   ns.totalCost<double>();
934    /// \endcode
935    /// It is useful if the total cost cannot be stored in the \c Cost
936    /// type of the algorithm, which is the default return type of the
937    /// function.
938    ///
939    /// \pre \ref run() must be called before using this function.
940    template <typename Number>
941    Number totalCost() const {
942      Number c = 0;
943      for (ArcIt a(_graph); a != INVALID; ++a) {
944        int i = _arc_id[a];
945        c += Number(_flow[i]) * Number(_cost[i]);
946      }
947      return c;
948    }
949
950#ifndef DOXYGEN
951    Cost totalCost() const {
952      return totalCost<Cost>();
953    }
954#endif
955
956    /// \brief Return the flow on the given arc.
957    ///
958    /// This function returns the flow on the given arc.
959    ///
960    /// \pre \ref run() must be called before using this function.
961    Value flow(const Arc& a) const {
962      return _flow[_arc_id[a]];
963    }
964
965    /// \brief Return the flow map (the primal solution).
966    ///
967    /// This function copies the flow value on each arc into the given
968    /// map. The \c Value type of the algorithm must be convertible to
969    /// the \c Value type of the map.
970    ///
971    /// \pre \ref run() must be called before using this function.
972    template <typename FlowMap>
973    void flowMap(FlowMap &map) const {
974      for (ArcIt a(_graph); a != INVALID; ++a) {
975        map.set(a, _flow[_arc_id[a]]);
976      }
977    }
978
979    /// \brief Return the potential (dual value) of the given node.
980    ///
981    /// This function returns the potential (dual value) of the
982    /// given node.
983    ///
984    /// \pre \ref run() must be called before using this function.
985    Cost potential(const Node& n) const {
986      return _pi[_node_id[n]];
987    }
988
989    /// \brief Return the potential map (the dual solution).
990    ///
991    /// This function copies the potential (dual value) of each node
992    /// into the given map.
993    /// The \c Cost type of the algorithm must be convertible to the
994    /// \c Value type of the map.
995    ///
996    /// \pre \ref run() must be called before using this function.
997    template <typename PotentialMap>
998    void potentialMap(PotentialMap &map) const {
999      for (NodeIt n(_graph); n != INVALID; ++n) {
1000        map.set(n, _pi[_node_id[n]]);
1001      }
1002    }
1003
1004    /// @}
1005
1006  private:
1007
1008    // Initialize internal data structures
1009    bool init() {
1010      if (_node_num == 0) return false;
1011
1012      // Check the sum of supply values
1013      _sum_supply = 0;
1014      for (int i = 0; i != _node_num; ++i) {
1015        _sum_supply += _supply[i];
1016      }
1017      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1018             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1019
1020      // Remove non-zero lower bounds
1021      if (_have_lower) {
1022        for (int i = 0; i != _arc_num; ++i) {
1023          Value c = _lower[i];
1024          if (c >= 0) {
1025            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1026          } else {
1027            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1028          }
1029          _supply[_source[i]] -= c;
1030          _supply[_target[i]] += c;
1031        }
1032      } else {
1033        for (int i = 0; i != _arc_num; ++i) {
1034          _cap[i] = _upper[i];
1035        }
1036      }
1037
1038      // Initialize artifical cost
1039      Cost ART_COST;
1040      if (std::numeric_limits<Cost>::is_exact) {
1041        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1042      } else {
1043        ART_COST = std::numeric_limits<Cost>::min();
1044        for (int i = 0; i != _arc_num; ++i) {
1045          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1046        }
1047        ART_COST = (ART_COST + 1) * _node_num;
1048      }
1049
1050      // Initialize arc maps
1051      for (int i = 0; i != _arc_num; ++i) {
1052        _flow[i] = 0;
1053        _state[i] = STATE_LOWER;
1054      }
1055     
1056      // Set data for the artificial root node
1057      _root = _node_num;
1058      _parent[_root] = -1;
1059      _pred[_root] = -1;
1060      _thread[_root] = 0;
1061      _rev_thread[0] = _root;
1062      _succ_num[_root] = _node_num + 1;
1063      _last_succ[_root] = _root - 1;
1064      _supply[_root] = -_sum_supply;
1065      _pi[_root] = 0;
1066
1067      // Add artificial arcs and initialize the spanning tree data structure
1068      if (_sum_supply == 0) {
1069        // EQ supply constraints
1070        _search_arc_num = _arc_num;
1071        _all_arc_num = _arc_num + _node_num;
1072        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1073          _parent[u] = _root;
1074          _pred[u] = e;
1075          _thread[u] = u + 1;
1076          _rev_thread[u + 1] = u;
1077          _succ_num[u] = 1;
1078          _last_succ[u] = u;
1079          _cap[e] = INF;
1080          _state[e] = STATE_TREE;
1081          if (_supply[u] >= 0) {
1082            _forward[u] = true;
1083            _pi[u] = 0;
1084            _source[e] = u;
1085            _target[e] = _root;
1086            _flow[e] = _supply[u];
1087            _cost[e] = 0;
1088          } else {
1089            _forward[u] = false;
1090            _pi[u] = ART_COST;
1091            _source[e] = _root;
1092            _target[e] = u;
1093            _flow[e] = -_supply[u];
1094            _cost[e] = ART_COST;
1095          }
1096        }
1097      }
1098      else if (_sum_supply > 0) {
1099        // LEQ supply constraints
1100        _search_arc_num = _arc_num + _node_num;
1101        int f = _arc_num + _node_num;
1102        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1103          _parent[u] = _root;
1104          _thread[u] = u + 1;
1105          _rev_thread[u + 1] = u;
1106          _succ_num[u] = 1;
1107          _last_succ[u] = u;
1108          if (_supply[u] >= 0) {
1109            _forward[u] = true;
1110            _pi[u] = 0;
1111            _pred[u] = e;
1112            _source[e] = u;
1113            _target[e] = _root;
1114            _cap[e] = INF;
1115            _flow[e] = _supply[u];
1116            _cost[e] = 0;
1117            _state[e] = STATE_TREE;
1118          } else {
1119            _forward[u] = false;
1120            _pi[u] = ART_COST;
1121            _pred[u] = f;
1122            _source[f] = _root;
1123            _target[f] = u;
1124            _cap[f] = INF;
1125            _flow[f] = -_supply[u];
1126            _cost[f] = ART_COST;
1127            _state[f] = STATE_TREE;
1128            _source[e] = u;
1129            _target[e] = _root;
1130            _cap[e] = INF;
1131            _flow[e] = 0;
1132            _cost[e] = 0;
1133            _state[e] = STATE_LOWER;
1134            ++f;
1135          }
1136        }
1137        _all_arc_num = f;
1138      }
1139      else {
1140        // GEQ supply constraints
1141        _search_arc_num = _arc_num + _node_num;
1142        int f = _arc_num + _node_num;
1143        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1144          _parent[u] = _root;
1145          _thread[u] = u + 1;
1146          _rev_thread[u + 1] = u;
1147          _succ_num[u] = 1;
1148          _last_succ[u] = u;
1149          if (_supply[u] <= 0) {
1150            _forward[u] = false;
1151            _pi[u] = 0;
1152            _pred[u] = e;
1153            _source[e] = _root;
1154            _target[e] = u;
1155            _cap[e] = INF;
1156            _flow[e] = -_supply[u];
1157            _cost[e] = 0;
1158            _state[e] = STATE_TREE;
1159          } else {
1160            _forward[u] = true;
1161            _pi[u] = -ART_COST;
1162            _pred[u] = f;
1163            _source[f] = u;
1164            _target[f] = _root;
1165            _cap[f] = INF;
1166            _flow[f] = _supply[u];
1167            _state[f] = STATE_TREE;
1168            _cost[f] = ART_COST;
1169            _source[e] = _root;
1170            _target[e] = u;
1171            _cap[e] = INF;
1172            _flow[e] = 0;
1173            _cost[e] = 0;
1174            _state[e] = STATE_LOWER;
1175            ++f;
1176          }
1177        }
1178        _all_arc_num = f;
1179      }
1180
1181      return true;
1182    }
1183
1184    // Find the join node
1185    void findJoinNode() {
1186      int u = _source[in_arc];
1187      int v = _target[in_arc];
1188      while (u != v) {
1189        if (_succ_num[u] < _succ_num[v]) {
1190          u = _parent[u];
1191        } else {
1192          v = _parent[v];
1193        }
1194      }
1195      join = u;
1196    }
1197
1198    // Find the leaving arc of the cycle and returns true if the
1199    // leaving arc is not the same as the entering arc
1200    bool findLeavingArc() {
1201      // Initialize first and second nodes according to the direction
1202      // of the cycle
1203      if (_state[in_arc] == STATE_LOWER) {
1204        first  = _source[in_arc];
1205        second = _target[in_arc];
1206      } else {
1207        first  = _target[in_arc];
1208        second = _source[in_arc];
1209      }
1210      delta = _cap[in_arc];
1211      int result = 0;
1212      Value d;
1213      int e;
1214
1215      // Search the cycle along the path form the first node to the root
1216      for (int u = first; u != join; u = _parent[u]) {
1217        e = _pred[u];
1218        d = _forward[u] ?
1219          _flow[e] : (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]);
1220        if (d < delta) {
1221          delta = d;
1222          u_out = u;
1223          result = 1;
1224        }
1225      }
1226      // Search the cycle along the path form the second node to the root
1227      for (int u = second; u != join; u = _parent[u]) {
1228        e = _pred[u];
1229        d = _forward[u] ?
1230          (_cap[e] >= MAX ? INF : _cap[e] - _flow[e]) : _flow[e];
1231        if (d <= delta) {
1232          delta = d;
1233          u_out = u;
1234          result = 2;
1235        }
1236      }
1237
1238      if (result == 1) {
1239        u_in = first;
1240        v_in = second;
1241      } else {
1242        u_in = second;
1243        v_in = first;
1244      }
1245      return result != 0;
1246    }
1247
1248    // Change _flow and _state vectors
1249    void changeFlow(bool change) {
1250      // Augment along the cycle
1251      if (delta > 0) {
1252        Value val = _state[in_arc] * delta;
1253        _flow[in_arc] += val;
1254        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1255          _flow[_pred[u]] += _forward[u] ? -val : val;
1256        }
1257        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1258          _flow[_pred[u]] += _forward[u] ? val : -val;
1259        }
1260      }
1261      // Update the state of the entering and leaving arcs
1262      if (change) {
1263        _state[in_arc] = STATE_TREE;
1264        _state[_pred[u_out]] =
1265          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1266      } else {
1267        _state[in_arc] = -_state[in_arc];
1268      }
1269    }
1270
1271    // Update the tree structure
1272    void updateTreeStructure() {
1273      int u, w;
1274      int old_rev_thread = _rev_thread[u_out];
1275      int old_succ_num = _succ_num[u_out];
1276      int old_last_succ = _last_succ[u_out];
1277      v_out = _parent[u_out];
1278
1279      u = _last_succ[u_in];  // the last successor of u_in
1280      right = _thread[u];    // the node after it
1281
1282      // Handle the case when old_rev_thread equals to v_in
1283      // (it also means that join and v_out coincide)
1284      if (old_rev_thread == v_in) {
1285        last = _thread[_last_succ[u_out]];
1286      } else {
1287        last = _thread[v_in];
1288      }
1289
1290      // Update _thread and _parent along the stem nodes (i.e. the nodes
1291      // between u_in and u_out, whose parent have to be changed)
1292      _thread[v_in] = stem = u_in;
1293      _dirty_revs.clear();
1294      _dirty_revs.push_back(v_in);
1295      par_stem = v_in;
1296      while (stem != u_out) {
1297        // Insert the next stem node into the thread list
1298        new_stem = _parent[stem];
1299        _thread[u] = new_stem;
1300        _dirty_revs.push_back(u);
1301
1302        // Remove the subtree of stem from the thread list
1303        w = _rev_thread[stem];
1304        _thread[w] = right;
1305        _rev_thread[right] = w;
1306
1307        // Change the parent node and shift stem nodes
1308        _parent[stem] = par_stem;
1309        par_stem = stem;
1310        stem = new_stem;
1311
1312        // Update u and right
1313        u = _last_succ[stem] == _last_succ[par_stem] ?
1314          _rev_thread[par_stem] : _last_succ[stem];
1315        right = _thread[u];
1316      }
1317      _parent[u_out] = par_stem;
1318      _thread[u] = last;
1319      _rev_thread[last] = u;
1320      _last_succ[u_out] = u;
1321
1322      // Remove the subtree of u_out from the thread list except for
1323      // the case when old_rev_thread equals to v_in
1324      // (it also means that join and v_out coincide)
1325      if (old_rev_thread != v_in) {
1326        _thread[old_rev_thread] = right;
1327        _rev_thread[right] = old_rev_thread;
1328      }
1329
1330      // Update _rev_thread using the new _thread values
1331      for (int i = 0; i < int(_dirty_revs.size()); ++i) {
1332        u = _dirty_revs[i];
1333        _rev_thread[_thread[u]] = u;
1334      }
1335
1336      // Update _pred, _forward, _last_succ and _succ_num for the
1337      // stem nodes from u_out to u_in
1338      int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1339      u = u_out;
1340      while (u != u_in) {
1341        w = _parent[u];
1342        _pred[u] = _pred[w];
1343        _forward[u] = !_forward[w];
1344        tmp_sc += _succ_num[u] - _succ_num[w];
1345        _succ_num[u] = tmp_sc;
1346        _last_succ[w] = tmp_ls;
1347        u = w;
1348      }
1349      _pred[u_in] = in_arc;
1350      _forward[u_in] = (u_in == _source[in_arc]);
1351      _succ_num[u_in] = old_succ_num;
1352
1353      // Set limits for updating _last_succ form v_in and v_out
1354      // towards the root
1355      int up_limit_in = -1;
1356      int up_limit_out = -1;
1357      if (_last_succ[join] == v_in) {
1358        up_limit_out = join;
1359      } else {
1360        up_limit_in = join;
1361      }
1362
1363      // Update _last_succ from v_in towards the root
1364      for (u = v_in; u != up_limit_in && _last_succ[u] == v_in;
1365           u = _parent[u]) {
1366        _last_succ[u] = _last_succ[u_out];
1367      }
1368      // Update _last_succ from v_out towards the root
1369      if (join != old_rev_thread && v_in != old_rev_thread) {
1370        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1371             u = _parent[u]) {
1372          _last_succ[u] = old_rev_thread;
1373        }
1374      } else {
1375        for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1376             u = _parent[u]) {
1377          _last_succ[u] = _last_succ[u_out];
1378        }
1379      }
1380
1381      // Update _succ_num from v_in to join
1382      for (u = v_in; u != join; u = _parent[u]) {
1383        _succ_num[u] += old_succ_num;
1384      }
1385      // Update _succ_num from v_out to join
1386      for (u = v_out; u != join; u = _parent[u]) {
1387        _succ_num[u] -= old_succ_num;
1388      }
1389    }
1390
1391    // Update potentials
1392    void updatePotential() {
1393      Cost sigma = _forward[u_in] ?
1394        _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] :
1395        _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]];
1396      // Update potentials in the subtree, which has been moved
1397      int end = _thread[_last_succ[u_in]];
1398      for (int u = u_in; u != end; u = _thread[u]) {
1399        _pi[u] += sigma;
1400      }
1401    }
1402
1403    // Execute the algorithm
1404    ProblemType start(PivotRule pivot_rule) {
1405      // Select the pivot rule implementation
1406      switch (pivot_rule) {
1407        case FIRST_ELIGIBLE:
1408          return start<FirstEligiblePivotRule>();
1409        case BEST_ELIGIBLE:
1410          return start<BestEligiblePivotRule>();
1411        case BLOCK_SEARCH:
1412          return start<BlockSearchPivotRule>();
1413        case CANDIDATE_LIST:
1414          return start<CandidateListPivotRule>();
1415        case ALTERING_LIST:
1416          return start<AlteringListPivotRule>();
1417      }
1418      return INFEASIBLE; // avoid warning
1419    }
1420
1421    template <typename PivotRuleImpl>
1422    ProblemType start() {
1423      PivotRuleImpl pivot(*this);
1424
1425      // Execute the Network Simplex algorithm
1426      while (pivot.findEnteringArc()) {
1427        findJoinNode();
1428        bool change = findLeavingArc();
1429        if (delta >= MAX) return UNBOUNDED;
1430        changeFlow(change);
1431        if (change) {
1432          updateTreeStructure();
1433          updatePotential();
1434        }
1435      }
1436     
1437      // Check feasibility
1438      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1439        if (_flow[e] != 0) return INFEASIBLE;
1440      }
1441
1442      // Transform the solution and the supply map to the original form
1443      if (_have_lower) {
1444        for (int i = 0; i != _arc_num; ++i) {
1445          Value c = _lower[i];
1446          if (c != 0) {
1447            _flow[i] += c;
1448            _supply[_source[i]] += c;
1449            _supply[_target[i]] -= c;
1450          }
1451        }
1452      }
1453     
1454      // Shift potentials to meet the requirements of the GEQ/LEQ type
1455      // optimality conditions
1456      if (_sum_supply == 0) {
1457        if (_stype == GEQ) {
1458          Cost max_pot = std::numeric_limits<Cost>::min();
1459          for (int i = 0; i != _node_num; ++i) {
1460            if (_pi[i] > max_pot) max_pot = _pi[i];
1461          }
1462          if (max_pot > 0) {
1463            for (int i = 0; i != _node_num; ++i)
1464              _pi[i] -= max_pot;
1465          }
1466        } else {
1467          Cost min_pot = std::numeric_limits<Cost>::max();
1468          for (int i = 0; i != _node_num; ++i) {
1469            if (_pi[i] < min_pot) min_pot = _pi[i];
1470          }
1471          if (min_pot < 0) {
1472            for (int i = 0; i != _node_num; ++i)
1473              _pi[i] -= min_pot;
1474          }
1475        }
1476      }
1477
1478      return OPTIMAL;
1479    }
1480
1481  }; //class NetworkSimplex
1482
1483  ///@}
1484
1485} //namespace lemon
1486
1487#endif //LEMON_NETWORK_SIMPLEX_H
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