# source:lemon/lemon/network_simplex.h@1047:ddd3c0d3d9bf

Last change on this file since 1047:ddd3c0d3d9bf was 1026:9312d6c89d02, checked in by Alpar Juttner <alpar@…>, 9 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 *
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
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, \ref NetworkSimplex and \ref CostScaling are the fastest
51  /// implementations available in LEMON for this problem.
52  /// Furthermore, this class supports both directions of the supply/demand
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 \c V and \c C must be signed number types.
67  /// \warning All input data (capacities, supply values, and costs) must
68  /// be integer.
69  ///
70  /// \note %NetworkSimplex provides five different pivot rule
71  /// implementations, from which the most efficient one is used
73  template <typename GR, typename V = int, typename C = V>
74  class NetworkSimplex
75  {
76  public:
77
78    /// The type of the flow amounts, capacity bounds and supply values
79    typedef V Value;
80    /// The type of the arc costs
81    typedef C Cost;
82
83  public:
84
85    /// \brief Problem type constants for the \c run() function.
86    ///
87    /// Enum type containing the problem type constants that can be
88    /// returned by the \ref run() function of the algorithm.
89    enum ProblemType {
90      /// The problem has no feasible solution (flow).
91      INFEASIBLE,
92      /// The problem has optimal solution (i.e. it is feasible and
93      /// bounded), and the algorithm has found optimal flow and node
94      /// potentials (primal and dual solutions).
95      OPTIMAL,
96      /// The objective function of the problem is unbounded, i.e.
97      /// there is a directed cycle having negative total cost and
98      /// infinite upper bound.
99      UNBOUNDED
100    };
101
102    /// \brief Constants for selecting the type of the supply constraints.
103    ///
104    /// Enum type containing constants for selecting the supply type,
105    /// i.e. the direction of the inequalities in the supply/demand
106    /// constraints of the \ref min_cost_flow "minimum cost flow problem".
107    ///
108    /// The default supply type is \c GEQ, the \c LEQ type can be
109    /// selected using \ref supplyType().
110    /// The equality form is a special case of both supply types.
111    enum SupplyType {
112      /// This option means that there are <em>"greater or equal"</em>
113      /// supply/demand constraints in the definition of the problem.
114      GEQ,
115      /// This option means that there are <em>"less or equal"</em>
116      /// supply/demand constraints in the definition of the problem.
117      LEQ
118    };
119
120    /// \brief Constants for selecting the pivot rule.
121    ///
122    /// Enum type containing constants for selecting the pivot rule for
123    /// the \ref run() function.
124    ///
125    /// \ref NetworkSimplex provides five different pivot rule
126    /// implementations that significantly affect the running time
127    /// of the algorithm.
128    /// By default, \ref BLOCK_SEARCH "Block Search" is used, which
129    /// turend out to be the most efficient and the most robust on various
130    /// test inputs.
131    /// However, another pivot rule can be selected using the \ref run()
132    /// function with the proper parameter.
133    enum PivotRule {
134
135      /// The \e First \e Eligible pivot rule.
136      /// The next eligible arc is selected in a wraparound fashion
137      /// in every iteration.
138      FIRST_ELIGIBLE,
139
140      /// The \e Best \e Eligible pivot rule.
141      /// The best eligible arc is selected in every iteration.
142      BEST_ELIGIBLE,
143
144      /// The \e Block \e Search pivot rule.
145      /// A specified number of arcs are examined in every iteration
146      /// in a wraparound fashion and the best eligible arc is selected
147      /// from this block.
148      BLOCK_SEARCH,
149
150      /// The \e Candidate \e List pivot rule.
151      /// In a major iteration a candidate list is built from eligible arcs
152      /// in a wraparound fashion and in the following minor iterations
153      /// the best eligible arc is selected from this list.
154      CANDIDATE_LIST,
155
156      /// The \e Altering \e Candidate \e List pivot rule.
157      /// It is a modified version of the Candidate List method.
158      /// It keeps only the several best eligible arcs from the former
159      /// candidate list and extends this list in every iteration.
160      ALTERING_LIST
161    };
162
163  private:
164
165    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
166
167    typedef std::vector<int> IntVector;
168    typedef std::vector<Value> ValueVector;
169    typedef std::vector<Cost> CostVector;
170    typedef std::vector<signed char> CharVector;
171    // Note: vector<signed char> is used instead of vector<ArcState> and
172    // vector<ArcDirection> for efficiency reasons
173
174    // State constants for arcs
175    enum ArcState {
176      STATE_UPPER = -1,
177      STATE_TREE  =  0,
178      STATE_LOWER =  1
179    };
180
181    // Direction constants for tree arcs
182    enum ArcDirection {
183      DIR_DOWN = -1,
184      DIR_UP   =  1
185    };
186
187  private:
188
189    // Data related to the underlying digraph
190    const GR &_graph;
191    int _node_num;
192    int _arc_num;
193    int _all_arc_num;
194    int _search_arc_num;
195
196    // Parameters of the problem
197    bool _have_lower;
198    SupplyType _stype;
199    Value _sum_supply;
200
201    // Data structures for storing the digraph
202    IntNodeMap _node_id;
203    IntArcMap _arc_id;
204    IntVector _source;
205    IntVector _target;
206    bool _arc_mixing;
207
208    // Node and arc data
209    ValueVector _lower;
210    ValueVector _upper;
211    ValueVector _cap;
212    CostVector _cost;
213    ValueVector _supply;
214    ValueVector _flow;
215    CostVector _pi;
216
217    // Data for storing the spanning tree structure
218    IntVector _parent;
219    IntVector _pred;
222    IntVector _succ_num;
223    IntVector _last_succ;
224    CharVector _pred_dir;
225    CharVector _state;
226    IntVector _dirty_revs;
227    int _root;
228
229    // Temporary data used in the current pivot iteration
230    int in_arc, join, u_in, v_in, u_out, v_out;
231    Value delta;
232
233    const Value MAX;
234
235  public:
236
237    /// \brief Constant for infinite upper bounds (capacities).
238    ///
239    /// Constant for infinite upper bounds (capacities).
240    /// It is \c std::numeric_limits<Value>::infinity() if available,
241    /// \c std::numeric_limits<Value>::max() otherwise.
242    const Value INF;
243
244  private:
245
246    // Implementation of the First Eligible pivot rule
247    class FirstEligiblePivotRule
248    {
249    private:
250
251      // References to the NetworkSimplex class
252      const IntVector  &_source;
253      const IntVector  &_target;
254      const CostVector &_cost;
255      const CharVector &_state;
256      const CostVector &_pi;
257      int &_in_arc;
258      int _search_arc_num;
259
260      // Pivot rule data
261      int _next_arc;
262
263    public:
264
265      // Constructor
266      FirstEligiblePivotRule(NetworkSimplex &ns) :
267        _source(ns._source), _target(ns._target),
268        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
269        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
270        _next_arc(0)
271      {}
272
273      // Find next entering arc
274      bool findEnteringArc() {
275        Cost c;
276        for (int e = _next_arc; e != _search_arc_num; ++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        for (int e = 0; e != _next_arc; ++e) {
285          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
286          if (c < 0) {
287            _in_arc = e;
288            _next_arc = e + 1;
289            return true;
290          }
291        }
292        return false;
293      }
294
295    }; //class FirstEligiblePivotRule
296
297
298    // Implementation of the Best Eligible pivot rule
299    class BestEligiblePivotRule
300    {
301    private:
302
303      // References to the NetworkSimplex class
304      const IntVector  &_source;
305      const IntVector  &_target;
306      const CostVector &_cost;
307      const CharVector &_state;
308      const CostVector &_pi;
309      int &_in_arc;
310      int _search_arc_num;
311
312    public:
313
314      // Constructor
315      BestEligiblePivotRule(NetworkSimplex &ns) :
316        _source(ns._source), _target(ns._target),
317        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
318        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num)
319      {}
320
321      // Find next entering arc
322      bool findEnteringArc() {
323        Cost c, min = 0;
324        for (int e = 0; e != _search_arc_num; ++e) {
325          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
326          if (c < min) {
327            min = c;
328            _in_arc = e;
329          }
330        }
331        return min < 0;
332      }
333
334    }; //class BestEligiblePivotRule
335
336
337    // Implementation of the Block Search pivot rule
338    class BlockSearchPivotRule
339    {
340    private:
341
342      // References to the NetworkSimplex class
343      const IntVector  &_source;
344      const IntVector  &_target;
345      const CostVector &_cost;
346      const CharVector &_state;
347      const CostVector &_pi;
348      int &_in_arc;
349      int _search_arc_num;
350
351      // Pivot rule data
352      int _block_size;
353      int _next_arc;
354
355    public:
356
357      // Constructor
358      BlockSearchPivotRule(NetworkSimplex &ns) :
359        _source(ns._source), _target(ns._target),
360        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
361        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
362        _next_arc(0)
363      {
364        // The main parameters of the pivot rule
365        const double BLOCK_SIZE_FACTOR = 1.0;
366        const int MIN_BLOCK_SIZE = 10;
367
368        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
369                                    std::sqrt(double(_search_arc_num))),
370                                MIN_BLOCK_SIZE );
371      }
372
373      // Find next entering arc
374      bool findEnteringArc() {
375        Cost c, min = 0;
376        int cnt = _block_size;
377        int e;
378        for (e = _next_arc; e != _search_arc_num; ++e) {
379          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
380          if (c < min) {
381            min = c;
382            _in_arc = e;
383          }
384          if (--cnt == 0) {
385            if (min < 0) goto search_end;
386            cnt = _block_size;
387          }
388        }
389        for (e = 0; e != _next_arc; ++e) {
390          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
391          if (c < min) {
392            min = c;
393            _in_arc = e;
394          }
395          if (--cnt == 0) {
396            if (min < 0) goto search_end;
397            cnt = _block_size;
398          }
399        }
400        if (min >= 0) return false;
401
402      search_end:
403        _next_arc = e;
404        return true;
405      }
406
407    }; //class BlockSearchPivotRule
408
409
410    // Implementation of the Candidate List pivot rule
411    class CandidateListPivotRule
412    {
413    private:
414
415      // References to the NetworkSimplex class
416      const IntVector  &_source;
417      const IntVector  &_target;
418      const CostVector &_cost;
419      const CharVector &_state;
420      const CostVector &_pi;
421      int &_in_arc;
422      int _search_arc_num;
423
424      // Pivot rule data
425      IntVector _candidates;
426      int _list_length, _minor_limit;
427      int _curr_length, _minor_count;
428      int _next_arc;
429
430    public:
431
432      /// Constructor
433      CandidateListPivotRule(NetworkSimplex &ns) :
434        _source(ns._source), _target(ns._target),
435        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
436        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
437        _next_arc(0)
438      {
439        // The main parameters of the pivot rule
440        const double LIST_LENGTH_FACTOR = 0.25;
441        const int MIN_LIST_LENGTH = 10;
442        const double MINOR_LIMIT_FACTOR = 0.1;
443        const int MIN_MINOR_LIMIT = 3;
444
445        _list_length = std::max( int(LIST_LENGTH_FACTOR *
446                                     std::sqrt(double(_search_arc_num))),
447                                 MIN_LIST_LENGTH );
448        _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length),
449                                 MIN_MINOR_LIMIT );
450        _curr_length = _minor_count = 0;
451        _candidates.resize(_list_length);
452      }
453
454      /// Find next entering arc
455      bool findEnteringArc() {
456        Cost min, c;
457        int e;
458        if (_curr_length > 0 && _minor_count < _minor_limit) {
459          // Minor iteration: select the best eligible arc from the
460          // current candidate list
461          ++_minor_count;
462          min = 0;
463          for (int i = 0; i < _curr_length; ++i) {
464            e = _candidates[i];
465            c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
466            if (c < min) {
467              min = c;
468              _in_arc = e;
469            }
470            else if (c >= 0) {
471              _candidates[i--] = _candidates[--_curr_length];
472            }
473          }
474          if (min < 0) return true;
475        }
476
477        // Major iteration: build a new candidate list
478        min = 0;
479        _curr_length = 0;
480        for (e = _next_arc; e != _search_arc_num; ++e) {
481          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
482          if (c < 0) {
483            _candidates[_curr_length++] = e;
484            if (c < min) {
485              min = c;
486              _in_arc = e;
487            }
488            if (_curr_length == _list_length) goto search_end;
489          }
490        }
491        for (e = 0; e != _next_arc; ++e) {
492          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
493          if (c < 0) {
494            _candidates[_curr_length++] = e;
495            if (c < min) {
496              min = c;
497              _in_arc = e;
498            }
499            if (_curr_length == _list_length) goto search_end;
500          }
501        }
502        if (_curr_length == 0) return false;
503
504      search_end:
505        _minor_count = 1;
506        _next_arc = e;
507        return true;
508      }
509
510    }; //class CandidateListPivotRule
511
512
513    // Implementation of the Altering Candidate List pivot rule
514    class AlteringListPivotRule
515    {
516    private:
517
518      // References to the NetworkSimplex class
519      const IntVector  &_source;
520      const IntVector  &_target;
521      const CostVector &_cost;
522      const CharVector &_state;
523      const CostVector &_pi;
524      int &_in_arc;
525      int _search_arc_num;
526
527      // Pivot rule data
529      int _next_arc;
530      IntVector _candidates;
531      CostVector _cand_cost;
532
533      // Functor class to compare arcs during sort of the candidate list
534      class SortFunc
535      {
536      private:
537        const CostVector &_map;
538      public:
539        SortFunc(const CostVector &map) : _map(map) {}
540        bool operator()(int left, int right) {
541          return _map[left] > _map[right];
542        }
543      };
544
545      SortFunc _sort_func;
546
547    public:
548
549      // Constructor
550      AlteringListPivotRule(NetworkSimplex &ns) :
551        _source(ns._source), _target(ns._target),
552        _cost(ns._cost), _state(ns._state), _pi(ns._pi),
553        _in_arc(ns.in_arc), _search_arc_num(ns._search_arc_num),
554        _next_arc(0), _cand_cost(ns._search_arc_num), _sort_func(_cand_cost)
555      {
556        // The main parameters of the pivot rule
557        const double BLOCK_SIZE_FACTOR = 1.0;
558        const int MIN_BLOCK_SIZE = 10;
559        const double HEAD_LENGTH_FACTOR = 0.1;
560        const int MIN_HEAD_LENGTH = 3;
561
562        _block_size = std::max( int(BLOCK_SIZE_FACTOR *
563                                    std::sqrt(double(_search_arc_num))),
564                                MIN_BLOCK_SIZE );
568        _curr_length = 0;
569      }
570
571      // Find next entering arc
572      bool findEnteringArc() {
573        // Check the current candidate list
574        int e;
575        Cost c;
576        for (int i = 0; i != _curr_length; ++i) {
577          e = _candidates[i];
578          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
579          if (c < 0) {
580            _cand_cost[e] = c;
581          } else {
582            _candidates[i--] = _candidates[--_curr_length];
583          }
584        }
585
586        // Extend the list
587        int cnt = _block_size;
589
590        for (e = _next_arc; e != _search_arc_num; ++e) {
591          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
592          if (c < 0) {
593            _cand_cost[e] = c;
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        for (e = 0; e != _next_arc; ++e) {
603          _cand_cost[e] = _state[e] *
604            (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);
605          if (_cand_cost[e] < 0) {
606            _candidates[_curr_length++] = e;
607          }
608          if (--cnt == 0) {
609            if (_curr_length > limit) goto search_end;
610            limit = 0;
611            cnt = _block_size;
612          }
613        }
614        if (_curr_length == 0) return false;
615
616      search_end:
617
618        // Make heap of the candidate list (approximating a partial sort)
619        make_heap( _candidates.begin(), _candidates.begin() + _curr_length,
620                   _sort_func );
621
622        // Pop the first element of the heap
623        _in_arc = _candidates[0];
624        _next_arc = e;
625        pop_heap( _candidates.begin(), _candidates.begin() + _curr_length,
626                  _sort_func );
627        _curr_length = std::min(_head_length, _curr_length - 1);
628        return true;
629      }
630
631    }; //class AlteringListPivotRule
632
633  public:
634
635    /// \brief Constructor.
636    ///
637    /// The constructor of the class.
638    ///
639    /// \param graph The digraph the algorithm runs on.
640    /// \param arc_mixing Indicate if the arcs will be stored in a
641    /// mixed order in the internal data structure.
642    /// In general, it leads to similar performance as using the original
643    /// arc order, but it makes the algorithm more robust and in special
644    /// cases, even significantly faster. Therefore, it is enabled by default.
645    NetworkSimplex(const GR& graph, bool arc_mixing = true) :
646      _graph(graph), _node_id(graph), _arc_id(graph),
647      _arc_mixing(arc_mixing),
648      MAX(std::numeric_limits<Value>::max()),
649      INF(std::numeric_limits<Value>::has_infinity ?
650          std::numeric_limits<Value>::infinity() : MAX)
651    {
652      // Check the number types
653      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
654        "The flow type of NetworkSimplex must be signed");
655      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
656        "The cost type of NetworkSimplex must be signed");
657
658      // Reset data structures
659      reset();
660    }
661
662    /// \name Parameters
663    /// The parameters of the algorithm can be specified using these
664    /// functions.
665
666    /// @{
667
668    /// \brief Set the lower bounds on the arcs.
669    ///
670    /// This function sets the lower bounds on the arcs.
671    /// If it is not used before calling \ref run(), the lower bounds
672    /// will be set to zero on all arcs.
673    ///
674    /// \param map An arc map storing the lower bounds.
675    /// Its \c Value type must be convertible to the \c Value type
676    /// of the algorithm.
677    ///
678    /// \return <tt>(*this)</tt>
679    template <typename LowerMap>
680    NetworkSimplex& lowerMap(const LowerMap& map) {
681      _have_lower = true;
682      for (ArcIt a(_graph); a != INVALID; ++a) {
683        _lower[_arc_id[a]] = map[a];
684      }
685      return *this;
686    }
687
688    /// \brief Set the upper bounds (capacities) on the arcs.
689    ///
690    /// This function sets the upper bounds (capacities) on the arcs.
691    /// If it is not used before calling \ref run(), the upper bounds
692    /// will be set to \ref INF on all arcs (i.e. the flow value will be
693    /// unbounded from above).
694    ///
695    /// \param map An arc map storing the upper bounds.
696    /// Its \c Value type must be convertible to the \c Value type
697    /// of the algorithm.
698    ///
699    /// \return <tt>(*this)</tt>
700    template<typename UpperMap>
701    NetworkSimplex& upperMap(const UpperMap& map) {
702      for (ArcIt a(_graph); a != INVALID; ++a) {
703        _upper[_arc_id[a]] = map[a];
704      }
705      return *this;
706    }
707
708    /// \brief Set the costs of the arcs.
709    ///
710    /// This function sets the costs of the arcs.
711    /// If it is not used before calling \ref run(), the costs
712    /// will be set to \c 1 on all arcs.
713    ///
714    /// \param map An arc map storing the costs.
715    /// Its \c Value type must be convertible to the \c Cost type
716    /// of the algorithm.
717    ///
718    /// \return <tt>(*this)</tt>
719    template<typename CostMap>
720    NetworkSimplex& costMap(const CostMap& map) {
721      for (ArcIt a(_graph); a != INVALID; ++a) {
722        _cost[_arc_id[a]] = map[a];
723      }
724      return *this;
725    }
726
727    /// \brief Set the supply values of the nodes.
728    ///
729    /// This function sets the supply values of the nodes.
730    /// If neither this function nor \ref stSupply() is used before
731    /// calling \ref run(), the supply of each node will be set to zero.
732    ///
733    /// \param map A node map storing the supply values.
734    /// Its \c Value type must be convertible to the \c Value type
735    /// of the algorithm.
736    ///
737    /// \return <tt>(*this)</tt>
738    ///
739    /// \sa supplyType()
740    template<typename SupplyMap>
741    NetworkSimplex& supplyMap(const SupplyMap& map) {
742      for (NodeIt n(_graph); n != INVALID; ++n) {
743        _supply[_node_id[n]] = map[n];
744      }
745      return *this;
746    }
747
748    /// \brief Set single source and target nodes and a supply value.
749    ///
750    /// This function sets a single source node and a single target node
751    /// and the required flow value.
752    /// If neither this function nor \ref supplyMap() is used before
753    /// calling \ref run(), the supply of each node will be set to zero.
754    ///
755    /// Using this function has the same effect as using \ref supplyMap()
756    /// with a map in which \c k is assigned to \c s, \c -k is
757    /// assigned to \c t and all other nodes have zero supply value.
758    ///
759    /// \param s The source node.
760    /// \param t The target node.
761    /// \param k The required amount of flow from node \c s to node \c t
762    /// (i.e. the supply of \c s and the demand of \c t).
763    ///
764    /// \return <tt>(*this)</tt>
765    NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) {
766      for (int i = 0; i != _node_num; ++i) {
767        _supply[i] = 0;
768      }
769      _supply[_node_id[s]] =  k;
770      _supply[_node_id[t]] = -k;
771      return *this;
772    }
773
774    /// \brief Set the type of the supply constraints.
775    ///
776    /// This function sets the type of the supply/demand constraints.
777    /// If it is not used before calling \ref run(), the \ref GEQ supply
778    /// type will be used.
779    ///
781    ///
782    /// \return <tt>(*this)</tt>
783    NetworkSimplex& supplyType(SupplyType supply_type) {
784      _stype = supply_type;
785      return *this;
786    }
787
788    /// @}
789
790    /// \name Execution Control
791    /// The algorithm can be executed using \ref run().
792
793    /// @{
794
795    /// \brief Run the algorithm.
796    ///
797    /// This function runs the algorithm.
798    /// The paramters can be specified using functions \ref lowerMap(),
799    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
800    /// \ref supplyType().
801    /// For example,
802    /// \code
803    ///   NetworkSimplex<ListDigraph> ns(graph);
804    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
805    ///     .supplyMap(sup).run();
806    /// \endcode
807    ///
808    /// This function can be called more than once. All the given parameters
809    /// are kept for the next call, unless \ref resetParams() or \ref reset()
810    /// is used, thus only the modified parameters have to be set again.
811    /// If the underlying digraph was also modified after the construction
812    /// of the class (or the last \ref reset() call), then the \ref reset()
813    /// function must be called.
814    ///
815    /// \param pivot_rule The pivot rule that will be used during the
817    ///
818    /// \return \c INFEASIBLE if no feasible flow exists,
819    /// \n \c OPTIMAL if the problem has optimal solution
820    /// (i.e. it is feasible and bounded), and the algorithm has found
821    /// optimal flow and node potentials (primal and dual solutions),
822    /// \n \c UNBOUNDED if the objective function of the problem is
823    /// unbounded, i.e. there is a directed cycle having negative total
824    /// cost and infinite upper bound.
825    ///
826    /// \see ProblemType, PivotRule
827    /// \see resetParams(), reset()
828    ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) {
829      if (!init()) return INFEASIBLE;
830      return start(pivot_rule);
831    }
832
833    /// \brief Reset all the parameters that have been given before.
834    ///
835    /// This function resets all the paramaters that have been given
836    /// before using functions \ref lowerMap(), \ref upperMap(),
837    /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType().
838    ///
839    /// It is useful for multiple \ref run() calls. Basically, all the given
840    /// parameters are kept for the next \ref run() call, unless
841    /// \ref resetParams() or \ref reset() is used.
842    /// If the underlying digraph was also modified after the construction
843    /// of the class or the last \ref reset() call, then the \ref reset()
844    /// function must be used, otherwise \ref resetParams() is sufficient.
845    ///
846    /// For example,
847    /// \code
848    ///   NetworkSimplex<ListDigraph> ns(graph);
849    ///
850    ///   // First run
851    ///   ns.lowerMap(lower).upperMap(upper).costMap(cost)
852    ///     .supplyMap(sup).run();
853    ///
854    ///   // Run again with modified cost map (resetParams() is not called,
855    ///   // so only the cost map have to be set again)
856    ///   cost[e] += 100;
857    ///   ns.costMap(cost).run();
858    ///
859    ///   // Run again from scratch using resetParams()
860    ///   // (the lower bounds will be set to zero on all arcs)
861    ///   ns.resetParams();
862    ///   ns.upperMap(capacity).costMap(cost)
863    ///     .supplyMap(sup).run();
864    /// \endcode
865    ///
866    /// \return <tt>(*this)</tt>
867    ///
868    /// \see reset(), run()
869    NetworkSimplex& resetParams() {
870      for (int i = 0; i != _node_num; ++i) {
871        _supply[i] = 0;
872      }
873      for (int i = 0; i != _arc_num; ++i) {
874        _lower[i] = 0;
875        _upper[i] = INF;
876        _cost[i] = 1;
877      }
878      _have_lower = false;
879      _stype = GEQ;
880      return *this;
881    }
882
883    /// \brief Reset the internal data structures and all the parameters
884    /// that have been given before.
885    ///
886    /// This function resets the internal data structures and all the
887    /// paramaters that have been given before using functions \ref lowerMap(),
888    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(),
889    /// \ref supplyType().
890    ///
891    /// It is useful for multiple \ref run() calls. Basically, all the given
892    /// parameters are kept for the next \ref run() call, unless
893    /// \ref resetParams() or \ref reset() is used.
894    /// If the underlying digraph was also modified after the construction
895    /// of the class or the last \ref reset() call, then the \ref reset()
896    /// function must be used, otherwise \ref resetParams() is sufficient.
897    ///
898    /// See \ref resetParams() for examples.
899    ///
900    /// \return <tt>(*this)</tt>
901    ///
902    /// \see resetParams(), run()
903    NetworkSimplex& reset() {
904      // Resize vectors
905      _node_num = countNodes(_graph);
906      _arc_num = countArcs(_graph);
907      int all_node_num = _node_num + 1;
908      int max_arc_num = _arc_num + 2 * _node_num;
909
910      _source.resize(max_arc_num);
911      _target.resize(max_arc_num);
912
913      _lower.resize(_arc_num);
914      _upper.resize(_arc_num);
915      _cap.resize(max_arc_num);
916      _cost.resize(max_arc_num);
917      _supply.resize(all_node_num);
918      _flow.resize(max_arc_num);
919      _pi.resize(all_node_num);
920
921      _parent.resize(all_node_num);
922      _pred.resize(all_node_num);
923      _pred_dir.resize(all_node_num);
926      _succ_num.resize(all_node_num);
927      _last_succ.resize(all_node_num);
928      _state.resize(max_arc_num);
929
930      // Copy the graph
931      int i = 0;
932      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
933        _node_id[n] = i;
934      }
935      if (_arc_mixing) {
936        // Store the arcs in a mixed order
937        const int skip = std::max(_arc_num / _node_num, 3);
938        int i = 0, j = 0;
939        for (ArcIt a(_graph); a != INVALID; ++a) {
940          _arc_id[a] = i;
941          _source[i] = _node_id[_graph.source(a)];
942          _target[i] = _node_id[_graph.target(a)];
943          if ((i += skip) >= _arc_num) i = ++j;
944        }
945      } else {
946        // Store the arcs in the original order
947        int i = 0;
948        for (ArcIt a(_graph); a != INVALID; ++a, ++i) {
949          _arc_id[a] = i;
950          _source[i] = _node_id[_graph.source(a)];
951          _target[i] = _node_id[_graph.target(a)];
952        }
953      }
954
955      // Reset parameters
956      resetParams();
957      return *this;
958    }
959
960    /// @}
961
962    /// \name Query Functions
963    /// The results of the algorithm can be obtained using these
964    /// functions.\n
965    /// The \ref run() function must be called before using them.
966
967    /// @{
968
969    /// \brief Return the total cost of the found flow.
970    ///
971    /// This function returns the total cost of the found flow.
972    /// Its complexity is O(e).
973    ///
974    /// \note The return type of the function can be specified as a
975    /// template parameter. For example,
976    /// \code
977    ///   ns.totalCost<double>();
978    /// \endcode
979    /// It is useful if the total cost cannot be stored in the \c Cost
980    /// type of the algorithm, which is the default return type of the
981    /// function.
982    ///
983    /// \pre \ref run() must be called before using this function.
984    template <typename Number>
985    Number totalCost() const {
986      Number c = 0;
987      for (ArcIt a(_graph); a != INVALID; ++a) {
988        int i = _arc_id[a];
989        c += Number(_flow[i]) * Number(_cost[i]);
990      }
991      return c;
992    }
993
994#ifndef DOXYGEN
995    Cost totalCost() const {
997    }
998#endif
999
1000    /// \brief Return the flow on the given arc.
1001    ///
1002    /// This function returns the flow on the given arc.
1003    ///
1004    /// \pre \ref run() must be called before using this function.
1005    Value flow(const Arc& a) const {
1006      return _flow[_arc_id[a]];
1007    }
1008
1009    /// \brief Return the flow map (the primal solution).
1010    ///
1011    /// This function copies the flow value on each arc into the given
1012    /// map. The \c Value type of the algorithm must be convertible to
1013    /// the \c Value type of the map.
1014    ///
1015    /// \pre \ref run() must be called before using this function.
1016    template <typename FlowMap>
1017    void flowMap(FlowMap &map) const {
1018      for (ArcIt a(_graph); a != INVALID; ++a) {
1019        map.set(a, _flow[_arc_id[a]]);
1020      }
1021    }
1022
1023    /// \brief Return the potential (dual value) of the given node.
1024    ///
1025    /// This function returns the potential (dual value) of the
1026    /// given node.
1027    ///
1028    /// \pre \ref run() must be called before using this function.
1029    Cost potential(const Node& n) const {
1030      return _pi[_node_id[n]];
1031    }
1032
1033    /// \brief Return the potential map (the dual solution).
1034    ///
1035    /// This function copies the potential (dual value) of each node
1036    /// into the given map.
1037    /// The \c Cost type of the algorithm must be convertible to the
1038    /// \c Value type of the map.
1039    ///
1040    /// \pre \ref run() must be called before using this function.
1041    template <typename PotentialMap>
1042    void potentialMap(PotentialMap &map) const {
1043      for (NodeIt n(_graph); n != INVALID; ++n) {
1044        map.set(n, _pi[_node_id[n]]);
1045      }
1046    }
1047
1048    /// @}
1049
1050  private:
1051
1052    // Initialize internal data structures
1053    bool init() {
1054      if (_node_num == 0) return false;
1055
1056      // Check the sum of supply values
1057      _sum_supply = 0;
1058      for (int i = 0; i != _node_num; ++i) {
1059        _sum_supply += _supply[i];
1060      }
1061      if ( !((_stype == GEQ && _sum_supply <= 0) ||
1062             (_stype == LEQ && _sum_supply >= 0)) ) return false;
1063
1064      // Remove non-zero lower bounds
1065      if (_have_lower) {
1066        for (int i = 0; i != _arc_num; ++i) {
1067          Value c = _lower[i];
1068          if (c >= 0) {
1069            _cap[i] = _upper[i] < MAX ? _upper[i] - c : INF;
1070          } else {
1071            _cap[i] = _upper[i] < MAX + c ? _upper[i] - c : INF;
1072          }
1073          _supply[_source[i]] -= c;
1074          _supply[_target[i]] += c;
1075        }
1076      } else {
1077        for (int i = 0; i != _arc_num; ++i) {
1078          _cap[i] = _upper[i];
1079        }
1080      }
1081
1082      // Initialize artifical cost
1083      Cost ART_COST;
1084      if (std::numeric_limits<Cost>::is_exact) {
1085        ART_COST = std::numeric_limits<Cost>::max() / 2 + 1;
1086      } else {
1087        ART_COST = 0;
1088        for (int i = 0; i != _arc_num; ++i) {
1089          if (_cost[i] > ART_COST) ART_COST = _cost[i];
1090        }
1091        ART_COST = (ART_COST + 1) * _node_num;
1092      }
1093
1094      // Initialize arc maps
1095      for (int i = 0; i != _arc_num; ++i) {
1096        _flow[i] = 0;
1097        _state[i] = STATE_LOWER;
1098      }
1099
1100      // Set data for the artificial root node
1101      _root = _node_num;
1102      _parent[_root] = -1;
1103      _pred[_root] = -1;
1106      _succ_num[_root] = _node_num + 1;
1107      _last_succ[_root] = _root - 1;
1108      _supply[_root] = -_sum_supply;
1109      _pi[_root] = 0;
1110
1111      // Add artificial arcs and initialize the spanning tree data structure
1112      if (_sum_supply == 0) {
1113        // EQ supply constraints
1114        _search_arc_num = _arc_num;
1115        _all_arc_num = _arc_num + _node_num;
1116        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1117          _parent[u] = _root;
1118          _pred[u] = e;
1119          _thread[u] = u + 1;
1120          _rev_thread[u + 1] = u;
1121          _succ_num[u] = 1;
1122          _last_succ[u] = u;
1123          _cap[e] = INF;
1124          _state[e] = STATE_TREE;
1125          if (_supply[u] >= 0) {
1126            _pred_dir[u] = DIR_UP;
1127            _pi[u] = 0;
1128            _source[e] = u;
1129            _target[e] = _root;
1130            _flow[e] = _supply[u];
1131            _cost[e] = 0;
1132          } else {
1133            _pred_dir[u] = DIR_DOWN;
1134            _pi[u] = ART_COST;
1135            _source[e] = _root;
1136            _target[e] = u;
1137            _flow[e] = -_supply[u];
1138            _cost[e] = ART_COST;
1139          }
1140        }
1141      }
1142      else if (_sum_supply > 0) {
1143        // LEQ supply constraints
1144        _search_arc_num = _arc_num + _node_num;
1145        int f = _arc_num + _node_num;
1146        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1147          _parent[u] = _root;
1148          _thread[u] = u + 1;
1149          _rev_thread[u + 1] = u;
1150          _succ_num[u] = 1;
1151          _last_succ[u] = u;
1152          if (_supply[u] >= 0) {
1153            _pred_dir[u] = DIR_UP;
1154            _pi[u] = 0;
1155            _pred[u] = e;
1156            _source[e] = u;
1157            _target[e] = _root;
1158            _cap[e] = INF;
1159            _flow[e] = _supply[u];
1160            _cost[e] = 0;
1161            _state[e] = STATE_TREE;
1162          } else {
1163            _pred_dir[u] = DIR_DOWN;
1164            _pi[u] = ART_COST;
1165            _pred[u] = f;
1166            _source[f] = _root;
1167            _target[f] = u;
1168            _cap[f] = INF;
1169            _flow[f] = -_supply[u];
1170            _cost[f] = ART_COST;
1171            _state[f] = STATE_TREE;
1172            _source[e] = u;
1173            _target[e] = _root;
1174            _cap[e] = INF;
1175            _flow[e] = 0;
1176            _cost[e] = 0;
1177            _state[e] = STATE_LOWER;
1178            ++f;
1179          }
1180        }
1181        _all_arc_num = f;
1182      }
1183      else {
1184        // GEQ supply constraints
1185        _search_arc_num = _arc_num + _node_num;
1186        int f = _arc_num + _node_num;
1187        for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) {
1188          _parent[u] = _root;
1189          _thread[u] = u + 1;
1190          _rev_thread[u + 1] = u;
1191          _succ_num[u] = 1;
1192          _last_succ[u] = u;
1193          if (_supply[u] <= 0) {
1194            _pred_dir[u] = DIR_DOWN;
1195            _pi[u] = 0;
1196            _pred[u] = e;
1197            _source[e] = _root;
1198            _target[e] = u;
1199            _cap[e] = INF;
1200            _flow[e] = -_supply[u];
1201            _cost[e] = 0;
1202            _state[e] = STATE_TREE;
1203          } else {
1204            _pred_dir[u] = DIR_UP;
1205            _pi[u] = -ART_COST;
1206            _pred[u] = f;
1207            _source[f] = u;
1208            _target[f] = _root;
1209            _cap[f] = INF;
1210            _flow[f] = _supply[u];
1211            _state[f] = STATE_TREE;
1212            _cost[f] = ART_COST;
1213            _source[e] = _root;
1214            _target[e] = u;
1215            _cap[e] = INF;
1216            _flow[e] = 0;
1217            _cost[e] = 0;
1218            _state[e] = STATE_LOWER;
1219            ++f;
1220          }
1221        }
1222        _all_arc_num = f;
1223      }
1224
1225      return true;
1226    }
1227
1228    // Find the join node
1229    void findJoinNode() {
1230      int u = _source[in_arc];
1231      int v = _target[in_arc];
1232      while (u != v) {
1233        if (_succ_num[u] < _succ_num[v]) {
1234          u = _parent[u];
1235        } else {
1236          v = _parent[v];
1237        }
1238      }
1239      join = u;
1240    }
1241
1242    // Find the leaving arc of the cycle and returns true if the
1243    // leaving arc is not the same as the entering arc
1244    bool findLeavingArc() {
1245      // Initialize first and second nodes according to the direction
1246      // of the cycle
1247      int first, second;
1248      if (_state[in_arc] == STATE_LOWER) {
1249        first  = _source[in_arc];
1250        second = _target[in_arc];
1251      } else {
1252        first  = _target[in_arc];
1253        second = _source[in_arc];
1254      }
1255      delta = _cap[in_arc];
1256      int result = 0;
1257      Value c, d;
1258      int e;
1259
1260      // Search the cycle form the first node to the join node
1261      for (int u = first; u != join; u = _parent[u]) {
1262        e = _pred[u];
1263        d = _flow[e];
1264        if (_pred_dir[u] == DIR_DOWN) {
1265          c = _cap[e];
1266          d = c >= MAX ? INF : c - d;
1267        }
1268        if (d < delta) {
1269          delta = d;
1270          u_out = u;
1271          result = 1;
1272        }
1273      }
1274
1275      // Search the cycle form the second node to the join node
1276      for (int u = second; u != join; u = _parent[u]) {
1277        e = _pred[u];
1278        d = _flow[e];
1279        if (_pred_dir[u] == DIR_UP) {
1280          c = _cap[e];
1281          d = c >= MAX ? INF : c - d;
1282        }
1283        if (d <= delta) {
1284          delta = d;
1285          u_out = u;
1286          result = 2;
1287        }
1288      }
1289
1290      if (result == 1) {
1291        u_in = first;
1292        v_in = second;
1293      } else {
1294        u_in = second;
1295        v_in = first;
1296      }
1297      return result != 0;
1298    }
1299
1300    // Change _flow and _state vectors
1301    void changeFlow(bool change) {
1302      // Augment along the cycle
1303      if (delta > 0) {
1304        Value val = _state[in_arc] * delta;
1305        _flow[in_arc] += val;
1306        for (int u = _source[in_arc]; u != join; u = _parent[u]) {
1307          _flow[_pred[u]] -= _pred_dir[u] * val;
1308        }
1309        for (int u = _target[in_arc]; u != join; u = _parent[u]) {
1310          _flow[_pred[u]] += _pred_dir[u] * val;
1311        }
1312      }
1313      // Update the state of the entering and leaving arcs
1314      if (change) {
1315        _state[in_arc] = STATE_TREE;
1316        _state[_pred[u_out]] =
1317          (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER;
1318      } else {
1319        _state[in_arc] = -_state[in_arc];
1320      }
1321    }
1322
1323    // Update the tree structure
1324    void updateTreeStructure() {
1326      int old_succ_num = _succ_num[u_out];
1327      int old_last_succ = _last_succ[u_out];
1328      v_out = _parent[u_out];
1329
1330      // Check if u_in and u_out coincide
1331      if (u_in == u_out) {
1332        // Update _parent, _pred, _pred_dir
1333        _parent[u_in] = v_in;
1334        _pred[u_in] = in_arc;
1335        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1336
1338        if (_thread[v_in] != u_out) {
1347        }
1348      } else {
1349        // Handle the case when old_rev_thread equals to v_in
1350        // (it also means that join and v_out coincide)
1353
1354        // Update _thread and _parent along the stem nodes (i.e. the nodes
1355        // between u_in and u_out, whose parent have to be changed)
1356        int stem = u_in;              // the current stem node
1357        int par_stem = v_in;          // the new parent of stem
1358        int next_stem;                // the next stem node
1359        int last = _last_succ[u_in];  // the last successor of stem
1360        int before, after = _thread[last];
1362        _dirty_revs.clear();
1363        _dirty_revs.push_back(v_in);
1364        while (stem != u_out) {
1365          // Insert the next stem node into the thread list
1366          next_stem = _parent[stem];
1368          _dirty_revs.push_back(last);
1369
1370          // Remove the subtree of stem from the thread list
1374
1375          // Change the parent node and shift stem nodes
1376          _parent[stem] = par_stem;
1377          par_stem = stem;
1378          stem = next_stem;
1379
1380          // Update last and after
1381          last = _last_succ[stem] == _last_succ[par_stem] ?
1384        }
1385        _parent[u_out] = par_stem;
1388        _last_succ[u_out] = last;
1389
1390        // Remove the subtree of u_out from the thread list except for
1391        // the case when old_rev_thread equals to v_in
1392        if (old_rev_thread != v_in) {
1395        }
1396
1398        for (int i = 0; i != int(_dirty_revs.size()); ++i) {
1399          int u = _dirty_revs[i];
1401        }
1402
1403        // Update _pred, _pred_dir, _last_succ and _succ_num for the
1404        // stem nodes from u_out to u_in
1405        int tmp_sc = 0, tmp_ls = _last_succ[u_out];
1406        for (int u = u_out, p = _parent[u]; u != u_in; u = p, p = _parent[u]) {
1407          _pred[u] = _pred[p];
1408          _pred_dir[u] = -_pred_dir[p];
1409          tmp_sc += _succ_num[u] - _succ_num[p];
1410          _succ_num[u] = tmp_sc;
1411          _last_succ[p] = tmp_ls;
1412        }
1413        _pred[u_in] = in_arc;
1414        _pred_dir[u_in] = u_in == _source[in_arc] ? DIR_UP : DIR_DOWN;
1415        _succ_num[u_in] = old_succ_num;
1416      }
1417
1418      // Update _last_succ from v_in towards the root
1419      int up_limit_out = _last_succ[join] == v_in ? join : -1;
1420      int last_succ_out = _last_succ[u_out];
1421      for (int u = v_in; u != -1 && _last_succ[u] == v_in; u = _parent[u]) {
1422        _last_succ[u] = last_succ_out;
1423      }
1424
1425      // Update _last_succ from v_out towards the root
1427        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1428             u = _parent[u]) {
1430        }
1431      }
1432      else if (last_succ_out != old_last_succ) {
1433        for (int u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ;
1434             u = _parent[u]) {
1435          _last_succ[u] = last_succ_out;
1436        }
1437      }
1438
1439      // Update _succ_num from v_in to join
1440      for (int u = v_in; u != join; u = _parent[u]) {
1441        _succ_num[u] += old_succ_num;
1442      }
1443      // Update _succ_num from v_out to join
1444      for (int u = v_out; u != join; u = _parent[u]) {
1445        _succ_num[u] -= old_succ_num;
1446      }
1447    }
1448
1449    // Update potentials in the subtree that has been moved
1450    void updatePotential() {
1451      Cost sigma = _pi[v_in] - _pi[u_in] -
1452                   _pred_dir[u_in] * _cost[in_arc];
1454      for (int u = u_in; u != end; u = _thread[u]) {
1455        _pi[u] += sigma;
1456      }
1457    }
1458
1459    // Heuristic initial pivots
1460    bool initialPivots() {
1461      Value curr, total = 0;
1462      std::vector<Node> supply_nodes, demand_nodes;
1463      for (NodeIt u(_graph); u != INVALID; ++u) {
1464        curr = _supply[_node_id[u]];
1465        if (curr > 0) {
1466          total += curr;
1467          supply_nodes.push_back(u);
1468        }
1469        else if (curr < 0) {
1470          demand_nodes.push_back(u);
1471        }
1472      }
1473      if (_sum_supply > 0) total -= _sum_supply;
1474      if (total <= 0) return true;
1475
1476      IntVector arc_vector;
1477      if (_sum_supply >= 0) {
1478        if (supply_nodes.size() == 1 && demand_nodes.size() == 1) {
1479          // Perform a reverse graph search from the sink to the source
1480          typename GR::template NodeMap<bool> reached(_graph, false);
1481          Node s = supply_nodes[0], t = demand_nodes[0];
1482          std::vector<Node> stack;
1483          reached[t] = true;
1484          stack.push_back(t);
1485          while (!stack.empty()) {
1486            Node u, v = stack.back();
1487            stack.pop_back();
1488            if (v == s) break;
1489            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1490              if (reached[u = _graph.source(a)]) continue;
1491              int j = _arc_id[a];
1492              if (_cap[j] >= total) {
1493                arc_vector.push_back(j);
1494                reached[u] = true;
1495                stack.push_back(u);
1496              }
1497            }
1498          }
1499        } else {
1500          // Find the min. cost incomming arc for each demand node
1501          for (int i = 0; i != int(demand_nodes.size()); ++i) {
1502            Node v = demand_nodes[i];
1503            Cost c, min_cost = std::numeric_limits<Cost>::max();
1504            Arc min_arc = INVALID;
1505            for (InArcIt a(_graph, v); a != INVALID; ++a) {
1506              c = _cost[_arc_id[a]];
1507              if (c < min_cost) {
1508                min_cost = c;
1509                min_arc = a;
1510              }
1511            }
1512            if (min_arc != INVALID) {
1513              arc_vector.push_back(_arc_id[min_arc]);
1514            }
1515          }
1516        }
1517      } else {
1518        // Find the min. cost outgoing arc for each supply node
1519        for (int i = 0; i != int(supply_nodes.size()); ++i) {
1520          Node u = supply_nodes[i];
1521          Cost c, min_cost = std::numeric_limits<Cost>::max();
1522          Arc min_arc = INVALID;
1523          for (OutArcIt a(_graph, u); a != INVALID; ++a) {
1524            c = _cost[_arc_id[a]];
1525            if (c < min_cost) {
1526              min_cost = c;
1527              min_arc = a;
1528            }
1529          }
1530          if (min_arc != INVALID) {
1531            arc_vector.push_back(_arc_id[min_arc]);
1532          }
1533        }
1534      }
1535
1536      // Perform heuristic initial pivots
1537      for (int i = 0; i != int(arc_vector.size()); ++i) {
1538        in_arc = arc_vector[i];
1539        if (_state[in_arc] * (_cost[in_arc] + _pi[_source[in_arc]] -
1540            _pi[_target[in_arc]]) >= 0) continue;
1541        findJoinNode();
1542        bool change = findLeavingArc();
1543        if (delta >= MAX) return false;
1544        changeFlow(change);
1545        if (change) {
1546          updateTreeStructure();
1547          updatePotential();
1548        }
1549      }
1550      return true;
1551    }
1552
1553    // Execute the algorithm
1554    ProblemType start(PivotRule pivot_rule) {
1555      // Select the pivot rule implementation
1556      switch (pivot_rule) {
1557        case FIRST_ELIGIBLE:
1558          return start<FirstEligiblePivotRule>();
1559        case BEST_ELIGIBLE:
1560          return start<BestEligiblePivotRule>();
1561        case BLOCK_SEARCH:
1562          return start<BlockSearchPivotRule>();
1563        case CANDIDATE_LIST:
1564          return start<CandidateListPivotRule>();
1565        case ALTERING_LIST:
1566          return start<AlteringListPivotRule>();
1567      }
1568      return INFEASIBLE; // avoid warning
1569    }
1570
1571    template <typename PivotRuleImpl>
1572    ProblemType start() {
1573      PivotRuleImpl pivot(*this);
1574
1575      // Perform heuristic initial pivots
1576      if (!initialPivots()) return UNBOUNDED;
1577
1578      // Execute the Network Simplex algorithm
1579      while (pivot.findEnteringArc()) {
1580        findJoinNode();
1581        bool change = findLeavingArc();
1582        if (delta >= MAX) return UNBOUNDED;
1583        changeFlow(change);
1584        if (change) {
1585          updateTreeStructure();
1586          updatePotential();
1587        }
1588      }
1589
1590      // Check feasibility
1591      for (int e = _search_arc_num; e != _all_arc_num; ++e) {
1592        if (_flow[e] != 0) return INFEASIBLE;
1593      }
1594
1595      // Transform the solution and the supply map to the original form
1596      if (_have_lower) {
1597        for (int i = 0; i != _arc_num; ++i) {
1598          Value c = _lower[i];
1599          if (c != 0) {
1600            _flow[i] += c;
1601            _supply[_source[i]] += c;
1602            _supply[_target[i]] -= c;
1603          }
1604        }
1605      }
1606
1607      // Shift potentials to meet the requirements of the GEQ/LEQ type
1608      // optimality conditions
1609      if (_sum_supply == 0) {
1610        if (_stype == GEQ) {
1611          Cost max_pot = -std::numeric_limits<Cost>::max();
1612          for (int i = 0; i != _node_num; ++i) {
1613            if (_pi[i] > max_pot) max_pot = _pi[i];
1614          }
1615          if (max_pot > 0) {
1616            for (int i = 0; i != _node_num; ++i)
1617              _pi[i] -= max_pot;
1618          }
1619        } else {
1620          Cost min_pot = std::numeric_limits<Cost>::max();
1621          for (int i = 0; i != _node_num; ++i) {
1622            if (_pi[i] < min_pot) min_pot = _pi[i];
1623          }
1624          if (min_pot < 0) {
1625            for (int i = 0; i != _node_num; ++i)
1626              _pi[i] -= min_pot;
1627          }
1628        }
1629      }
1630
1631      return OPTIMAL;
1632    }
1633
1634  }; //class NetworkSimplex
1635
1636  ///@}
1637
1638} //namespace lemon
1639
1640#endif //LEMON_NETWORK_SIMPLEX_H
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