COIN-OR::LEMON - Graph Library

source: lemon-main/lemon/cycle_canceling.h

Last change on this file was 1104:a78e5b779b69, checked in by Alpar Juttner <alpar@…>, 6 years ago

Merge bugfix #478

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1/* -*- mode: C++; indent-tabs-mode: nil; -*-
2 *
3 * This file is a part of LEMON, a generic C++ optimization library.
4 *
5 * Copyright (C) 2003-2013
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_CYCLE_CANCELING_H
20#define LEMON_CYCLE_CANCELING_H
21
22/// \ingroup min_cost_flow_algs
23/// \file
24/// \brief Cycle-canceling algorithms for finding a minimum cost flow.
25
26#include <vector>
27#include <limits>
28
29#include <lemon/core.h>
30#include <lemon/maps.h>
31#include <lemon/path.h>
32#include <lemon/math.h>
33#include <lemon/static_graph.h>
34#include <lemon/adaptors.h>
35#include <lemon/circulation.h>
36#include <lemon/bellman_ford.h>
37#include <lemon/howard_mmc.h>
38#include <lemon/hartmann_orlin_mmc.h>
39
40namespace lemon {
41
42  /// \addtogroup min_cost_flow_algs
43  /// @{
44
45  /// \brief Implementation of cycle-canceling algorithms for
46  /// finding a \ref min_cost_flow "minimum cost flow".
47  ///
48  /// \ref CycleCanceling implements three different cycle-canceling
49  /// algorithms for finding a \ref min_cost_flow "minimum cost flow"
50  /// \cite amo93networkflows, \cite klein67primal,
51  /// \cite goldberg89cyclecanceling.
52  /// The most efficent one is the \ref CANCEL_AND_TIGHTEN
53  /// "Cancel-and-Tighten" algorithm, thus it is the default method.
54  /// It runs in strongly polynomial time \f$O(n^2 m^2 \log n)\f$,
55  /// but in practice, it is typically orders of magnitude slower than
56  /// the scaling algorithms and \ref NetworkSimplex.
57  /// (For more information, see \ref min_cost_flow_algs "the module page".)
58  ///
59  /// Most of the parameters of the problem (except for the digraph)
60  /// can be given using separate functions, and the algorithm can be
61  /// executed using the \ref run() function. If some parameters are not
62  /// specified, then default values will be used.
63  ///
64  /// \tparam GR The digraph type the algorithm runs on.
65  /// \tparam V The number type used for flow amounts, capacity bounds
66  /// and supply values in the algorithm. By default, it is \c int.
67  /// \tparam C The number type used for costs and potentials in the
68  /// algorithm. By default, it is the same as \c V.
69  ///
70  /// \warning Both \c V and \c C must be signed number types.
71  /// \warning All input data (capacities, supply values, and costs) must
72  /// be integer.
73  /// \warning This algorithm does not support negative costs for
74  /// arcs having infinite upper bound.
75  ///
76  /// \note For more information about the three available methods,
77  /// see \ref Method.
78#ifdef DOXYGEN
79  template <typename GR, typename V, typename C>
80#else
81  template <typename GR, typename V = int, typename C = V>
82#endif
83  class CycleCanceling
84  {
85  public:
86
87    /// The type of the digraph
88    typedef GR Digraph;
89    /// The type of the flow amounts, capacity bounds and supply values
90    typedef V Value;
91    /// The type of the arc costs
92    typedef C Cost;
93
94  public:
95
96    /// \brief Problem type constants for the \c run() function.
97    ///
98    /// Enum type containing the problem type constants that can be
99    /// returned by the \ref run() function of the algorithm.
100    enum ProblemType {
101      /// The problem has no feasible solution (flow).
102      INFEASIBLE,
103      /// The problem has optimal solution (i.e. it is feasible and
104      /// bounded), and the algorithm has found optimal flow and node
105      /// potentials (primal and dual solutions).
106      OPTIMAL,
107      /// The digraph contains an arc of negative cost and infinite
108      /// upper bound. It means that the objective function is unbounded
109      /// on that arc, however, note that it could actually be bounded
110      /// over the feasible flows, but this algroithm cannot handle
111      /// these cases.
112      UNBOUNDED
113    };
114
115    /// \brief Constants for selecting the used method.
116    ///
117    /// Enum type containing constants for selecting the used method
118    /// for the \ref run() function.
119    ///
120    /// \ref CycleCanceling provides three different cycle-canceling
121    /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten"
122    /// is used, which is by far the most efficient and the most robust.
123    /// However, the other methods can be selected using the \ref run()
124    /// function with the proper parameter.
125    enum Method {
126      /// A simple cycle-canceling method, which uses the
127      /// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative
128      /// cycles in the residual network.
129      /// The number of Bellman-Ford iterations is bounded by a successively
130      /// increased limit.
131      SIMPLE_CYCLE_CANCELING,
132      /// The "Minimum Mean Cycle-Canceling" algorithm, which is a
133      /// well-known strongly polynomial method
134      /// \cite goldberg89cyclecanceling. It improves along a
135      /// \ref min_mean_cycle "minimum mean cycle" in each iteration.
136      /// Its running time complexity is \f$O(n^2 m^3 \log n)\f$.
137      MINIMUM_MEAN_CYCLE_CANCELING,
138      /// The "Cancel-and-Tighten" algorithm, which can be viewed as an
139      /// improved version of the previous method
140      /// \cite goldberg89cyclecanceling.
141      /// It is faster both in theory and in practice, its running time
142      /// complexity is \f$O(n^2 m^2 \log n)\f$.
143      CANCEL_AND_TIGHTEN
144    };
145
146  private:
147
148    TEMPLATE_DIGRAPH_TYPEDEFS(GR);
149
150    typedef std::vector<int> IntVector;
151    typedef std::vector<double> DoubleVector;
152    typedef std::vector<Value> ValueVector;
153    typedef std::vector<Cost> CostVector;
154    typedef std::vector<char> BoolVector;
155    // Note: vector<char> is used instead of vector<bool> for efficiency reasons
156
157  private:
158
159    template <typename KT, typename VT>
160    class StaticVectorMap {
161    public:
162      typedef KT Key;
163      typedef VT Value;
164
165      StaticVectorMap(std::vector<Value>& v) : _v(v) {}
166
167      const Value& operator[](const Key& key) const {
168        return _v[StaticDigraph::id(key)];
169      }
170
171      Value& operator[](const Key& key) {
172        return _v[StaticDigraph::id(key)];
173      }
174
175      void set(const Key& key, const Value& val) {
176        _v[StaticDigraph::id(key)] = val;
177      }
178
179    private:
180      std::vector<Value>& _v;
181    };
182
183    typedef StaticVectorMap<StaticDigraph::Node, Cost> CostNodeMap;
184    typedef StaticVectorMap<StaticDigraph::Arc, Cost> CostArcMap;
185
186  private:
187
188
189    // Data related to the underlying digraph
190    const GR &_graph;
191    int _node_num;
192    int _arc_num;
193    int _res_node_num;
194    int _res_arc_num;
195    int _root;
196
197    // Parameters of the problem
198    bool _has_lower;
199    Value _sum_supply;
200
201    // Data structures for storing the digraph
202    IntNodeMap _node_id;
203    IntArcMap _arc_idf;
204    IntArcMap _arc_idb;
205    IntVector _first_out;
206    BoolVector _forward;
207    IntVector _source;
208    IntVector _target;
209    IntVector _reverse;
210
211    // Node and arc data
212    ValueVector _lower;
213    ValueVector _upper;
214    CostVector _cost;
215    ValueVector _supply;
216
217    ValueVector _res_cap;
218    CostVector _pi;
219
220    // Data for a StaticDigraph structure
221    typedef std::pair<int, int> IntPair;
222    StaticDigraph _sgr;
223    std::vector<IntPair> _arc_vec;
224    std::vector<Cost> _cost_vec;
225    IntVector _id_vec;
226    CostArcMap _cost_map;
227    CostNodeMap _pi_map;
228
229  public:
230
231    /// \brief Constant for infinite upper bounds (capacities).
232    ///
233    /// Constant for infinite upper bounds (capacities).
234    /// It is \c std::numeric_limits<Value>::infinity() if available,
235    /// \c std::numeric_limits<Value>::max() otherwise.
236    const Value INF;
237
238  public:
239
240    /// \brief Constructor.
241    ///
242    /// The constructor of the class.
243    ///
244    /// \param graph The digraph the algorithm runs on.
245    CycleCanceling(const GR& graph) :
246      _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
247      _cost_map(_cost_vec), _pi_map(_pi),
248      INF(std::numeric_limits<Value>::has_infinity ?
249          std::numeric_limits<Value>::infinity() :
250          std::numeric_limits<Value>::max())
251    {
252      // Check the number types
253      LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
254        "The flow type of CycleCanceling must be signed");
255      LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
256        "The cost type of CycleCanceling must be signed");
257
258      // Reset data structures
259      reset();
260    }
261
262    /// \name Parameters
263    /// The parameters of the algorithm can be specified using these
264    /// functions.
265
266    /// @{
267
268    /// \brief Set the lower bounds on the arcs.
269    ///
270    /// This function sets the lower bounds on the arcs.
271    /// If it is not used before calling \ref run(), the lower bounds
272    /// will be set to zero on all arcs.
273    ///
274    /// \param map An arc map storing the lower bounds.
275    /// Its \c Value type must be convertible to the \c Value type
276    /// of the algorithm.
277    ///
278    /// \return <tt>(*this)</tt>
279    template <typename LowerMap>
280    CycleCanceling& lowerMap(const LowerMap& map) {
281      _has_lower = true;
282      for (ArcIt a(_graph); a != INVALID; ++a) {
283        _lower[_arc_idf[a]] = map[a];
284      }
285      return *this;
286    }
287
288    /// \brief Set the upper bounds (capacities) on the arcs.
289    ///
290    /// This function sets the upper bounds (capacities) on the arcs.
291    /// If it is not used before calling \ref run(), the upper bounds
292    /// will be set to \ref INF on all arcs (i.e. the flow value will be
293    /// unbounded from above).
294    ///
295    /// \param map An arc map storing the upper bounds.
296    /// Its \c Value type must be convertible to the \c Value type
297    /// of the algorithm.
298    ///
299    /// \return <tt>(*this)</tt>
300    template<typename UpperMap>
301    CycleCanceling& upperMap(const UpperMap& map) {
302      for (ArcIt a(_graph); a != INVALID; ++a) {
303        _upper[_arc_idf[a]] = map[a];
304      }
305      return *this;
306    }
307
308    /// \brief Set the costs of the arcs.
309    ///
310    /// This function sets the costs of the arcs.
311    /// If it is not used before calling \ref run(), the costs
312    /// will be set to \c 1 on all arcs.
313    ///
314    /// \param map An arc map storing the costs.
315    /// Its \c Value type must be convertible to the \c Cost type
316    /// of the algorithm.
317    ///
318    /// \return <tt>(*this)</tt>
319    template<typename CostMap>
320    CycleCanceling& costMap(const CostMap& map) {
321      for (ArcIt a(_graph); a != INVALID; ++a) {
322        _cost[_arc_idf[a]] =  map[a];
323        _cost[_arc_idb[a]] = -map[a];
324      }
325      return *this;
326    }
327
328    /// \brief Set the supply values of the nodes.
329    ///
330    /// This function sets the supply values of the nodes.
331    /// If neither this function nor \ref stSupply() is used before
332    /// calling \ref run(), the supply of each node will be set to zero.
333    ///
334    /// \param map A node map storing the supply values.
335    /// Its \c Value type must be convertible to the \c Value type
336    /// of the algorithm.
337    ///
338    /// \return <tt>(*this)</tt>
339    template<typename SupplyMap>
340    CycleCanceling& supplyMap(const SupplyMap& map) {
341      for (NodeIt n(_graph); n != INVALID; ++n) {
342        _supply[_node_id[n]] = map[n];
343      }
344      return *this;
345    }
346
347    /// \brief Set single source and target nodes and a supply value.
348    ///
349    /// This function sets a single source node and a single target node
350    /// and the required flow value.
351    /// If neither this function nor \ref supplyMap() is used before
352    /// calling \ref run(), the supply of each node will be set to zero.
353    ///
354    /// Using this function has the same effect as using \ref supplyMap()
355    /// with a map in which \c k is assigned to \c s, \c -k is
356    /// assigned to \c t and all other nodes have zero supply value.
357    ///
358    /// \param s The source node.
359    /// \param t The target node.
360    /// \param k The required amount of flow from node \c s to node \c t
361    /// (i.e. the supply of \c s and the demand of \c t).
362    ///
363    /// \return <tt>(*this)</tt>
364    CycleCanceling& stSupply(const Node& s, const Node& t, Value k) {
365      for (int i = 0; i != _res_node_num; ++i) {
366        _supply[i] = 0;
367      }
368      _supply[_node_id[s]] =  k;
369      _supply[_node_id[t]] = -k;
370      return *this;
371    }
372
373    /// @}
374
375    /// \name Execution control
376    /// The algorithm can be executed using \ref run().
377
378    /// @{
379
380    /// \brief Run the algorithm.
381    ///
382    /// This function runs the algorithm.
383    /// The paramters can be specified using functions \ref lowerMap(),
384    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
385    /// For example,
386    /// \code
387    ///   CycleCanceling<ListDigraph> cc(graph);
388    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
389    ///     .supplyMap(sup).run();
390    /// \endcode
391    ///
392    /// This function can be called more than once. All the given parameters
393    /// are kept for the next call, unless \ref resetParams() or \ref reset()
394    /// is used, thus only the modified parameters have to be set again.
395    /// If the underlying digraph was also modified after the construction
396    /// of the class (or the last \ref reset() call), then the \ref reset()
397    /// function must be called.
398    ///
399    /// \param method The cycle-canceling method that will be used.
400    /// For more information, see \ref Method.
401    ///
402    /// \return \c INFEASIBLE if no feasible flow exists,
403    /// \n \c OPTIMAL if the problem has optimal solution
404    /// (i.e. it is feasible and bounded), and the algorithm has found
405    /// optimal flow and node potentials (primal and dual solutions),
406    /// \n \c UNBOUNDED if the digraph contains an arc of negative cost
407    /// and infinite upper bound. It means that the objective function
408    /// is unbounded on that arc, however, note that it could actually be
409    /// bounded over the feasible flows, but this algroithm cannot handle
410    /// these cases.
411    ///
412    /// \see ProblemType, Method
413    /// \see resetParams(), reset()
414    ProblemType run(Method method = CANCEL_AND_TIGHTEN) {
415      ProblemType pt = init();
416      if (pt != OPTIMAL) return pt;
417      start(method);
418      return OPTIMAL;
419    }
420
421    /// \brief Reset all the parameters that have been given before.
422    ///
423    /// This function resets all the paramaters that have been given
424    /// before using functions \ref lowerMap(), \ref upperMap(),
425    /// \ref costMap(), \ref supplyMap(), \ref stSupply().
426    ///
427    /// It is useful for multiple \ref run() calls. Basically, all the given
428    /// parameters are kept for the next \ref run() call, unless
429    /// \ref resetParams() or \ref reset() is used.
430    /// If the underlying digraph was also modified after the construction
431    /// of the class or the last \ref reset() call, then the \ref reset()
432    /// function must be used, otherwise \ref resetParams() is sufficient.
433    ///
434    /// For example,
435    /// \code
436    ///   CycleCanceling<ListDigraph> cs(graph);
437    ///
438    ///   // First run
439    ///   cc.lowerMap(lower).upperMap(upper).costMap(cost)
440    ///     .supplyMap(sup).run();
441    ///
442    ///   // Run again with modified cost map (resetParams() is not called,
443    ///   // so only the cost map have to be set again)
444    ///   cost[e] += 100;
445    ///   cc.costMap(cost).run();
446    ///
447    ///   // Run again from scratch using resetParams()
448    ///   // (the lower bounds will be set to zero on all arcs)
449    ///   cc.resetParams();
450    ///   cc.upperMap(capacity).costMap(cost)
451    ///     .supplyMap(sup).run();
452    /// \endcode
453    ///
454    /// \return <tt>(*this)</tt>
455    ///
456    /// \see reset(), run()
457    CycleCanceling& resetParams() {
458      for (int i = 0; i != _res_node_num; ++i) {
459        _supply[i] = 0;
460      }
461      int limit = _first_out[_root];
462      for (int j = 0; j != limit; ++j) {
463        _lower[j] = 0;
464        _upper[j] = INF;
465        _cost[j] = _forward[j] ? 1 : -1;
466      }
467      for (int j = limit; j != _res_arc_num; ++j) {
468        _lower[j] = 0;
469        _upper[j] = INF;
470        _cost[j] = 0;
471        _cost[_reverse[j]] = 0;
472      }
473      _has_lower = false;
474      return *this;
475    }
476
477    /// \brief Reset the internal data structures and all the parameters
478    /// that have been given before.
479    ///
480    /// This function resets the internal data structures and all the
481    /// paramaters that have been given before using functions \ref lowerMap(),
482    /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply().
483    ///
484    /// It is useful for multiple \ref run() calls. Basically, all the given
485    /// parameters are kept for the next \ref run() call, unless
486    /// \ref resetParams() or \ref reset() is used.
487    /// If the underlying digraph was also modified after the construction
488    /// of the class or the last \ref reset() call, then the \ref reset()
489    /// function must be used, otherwise \ref resetParams() is sufficient.
490    ///
491    /// See \ref resetParams() for examples.
492    ///
493    /// \return <tt>(*this)</tt>
494    ///
495    /// \see resetParams(), run()
496    CycleCanceling& reset() {
497      // Resize vectors
498      _node_num = countNodes(_graph);
499      _arc_num = countArcs(_graph);
500      _res_node_num = _node_num + 1;
501      _res_arc_num = 2 * (_arc_num + _node_num);
502      _root = _node_num;
503
504      _first_out.resize(_res_node_num + 1);
505      _forward.resize(_res_arc_num);
506      _source.resize(_res_arc_num);
507      _target.resize(_res_arc_num);
508      _reverse.resize(_res_arc_num);
509
510      _lower.resize(_res_arc_num);
511      _upper.resize(_res_arc_num);
512      _cost.resize(_res_arc_num);
513      _supply.resize(_res_node_num);
514
515      _res_cap.resize(_res_arc_num);
516      _pi.resize(_res_node_num);
517
518      _arc_vec.reserve(_res_arc_num);
519      _cost_vec.reserve(_res_arc_num);
520      _id_vec.reserve(_res_arc_num);
521
522      // Copy the graph
523      int i = 0, j = 0, k = 2 * _arc_num + _node_num;
524      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
525        _node_id[n] = i;
526      }
527      i = 0;
528      for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
529        _first_out[i] = j;
530        for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
531          _arc_idf[a] = j;
532          _forward[j] = true;
533          _source[j] = i;
534          _target[j] = _node_id[_graph.runningNode(a)];
535        }
536        for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
537          _arc_idb[a] = j;
538          _forward[j] = false;
539          _source[j] = i;
540          _target[j] = _node_id[_graph.runningNode(a)];
541        }
542        _forward[j] = false;
543        _source[j] = i;
544        _target[j] = _root;
545        _reverse[j] = k;
546        _forward[k] = true;
547        _source[k] = _root;
548        _target[k] = i;
549        _reverse[k] = j;
550        ++j; ++k;
551      }
552      _first_out[i] = j;
553      _first_out[_res_node_num] = k;
554      for (ArcIt a(_graph); a != INVALID; ++a) {
555        int fi = _arc_idf[a];
556        int bi = _arc_idb[a];
557        _reverse[fi] = bi;
558        _reverse[bi] = fi;
559      }
560
561      // Reset parameters
562      resetParams();
563      return *this;
564    }
565
566    /// @}
567
568    /// \name Query Functions
569    /// The results of the algorithm can be obtained using these
570    /// functions.\n
571    /// The \ref run() function must be called before using them.
572
573    /// @{
574
575    /// \brief Return the total cost of the found flow.
576    ///
577    /// This function returns the total cost of the found flow.
578    /// Its complexity is O(m).
579    ///
580    /// \note The return type of the function can be specified as a
581    /// template parameter. For example,
582    /// \code
583    ///   cc.totalCost<double>();
584    /// \endcode
585    /// It is useful if the total cost cannot be stored in the \c Cost
586    /// type of the algorithm, which is the default return type of the
587    /// function.
588    ///
589    /// \pre \ref run() must be called before using this function.
590    template <typename Number>
591    Number totalCost() const {
592      Number c = 0;
593      for (ArcIt a(_graph); a != INVALID; ++a) {
594        int i = _arc_idb[a];
595        c += static_cast<Number>(_res_cap[i]) *
596             (-static_cast<Number>(_cost[i]));
597      }
598      return c;
599    }
600
601#ifndef DOXYGEN
602    Cost totalCost() const {
603      return totalCost<Cost>();
604    }
605#endif
606
607    /// \brief Return the flow on the given arc.
608    ///
609    /// This function returns the flow on the given arc.
610    ///
611    /// \pre \ref run() must be called before using this function.
612    Value flow(const Arc& a) const {
613      return _res_cap[_arc_idb[a]];
614    }
615
616    /// \brief Copy the flow values (the primal solution) into the
617    /// given map.
618    ///
619    /// This function copies the flow value on each arc into the given
620    /// map. The \c Value type of the algorithm must be convertible to
621    /// the \c Value type of the map.
622    ///
623    /// \pre \ref run() must be called before using this function.
624    template <typename FlowMap>
625    void flowMap(FlowMap &map) const {
626      for (ArcIt a(_graph); a != INVALID; ++a) {
627        map.set(a, _res_cap[_arc_idb[a]]);
628      }
629    }
630
631    /// \brief Return the potential (dual value) of the given node.
632    ///
633    /// This function returns the potential (dual value) of the
634    /// given node.
635    ///
636    /// \pre \ref run() must be called before using this function.
637    Cost potential(const Node& n) const {
638      return static_cast<Cost>(_pi[_node_id[n]]);
639    }
640
641    /// \brief Copy the potential values (the dual solution) into the
642    /// given map.
643    ///
644    /// This function copies the potential (dual value) of each node
645    /// into the given map.
646    /// The \c Cost type of the algorithm must be convertible to the
647    /// \c Value type of the map.
648    ///
649    /// \pre \ref run() must be called before using this function.
650    template <typename PotentialMap>
651    void potentialMap(PotentialMap &map) const {
652      for (NodeIt n(_graph); n != INVALID; ++n) {
653        map.set(n, static_cast<Cost>(_pi[_node_id[n]]));
654      }
655    }
656
657    /// @}
658
659  private:
660
661    // Initialize the algorithm
662    ProblemType init() {
663      if (_res_node_num <= 1) return INFEASIBLE;
664
665      // Check the sum of supply values
666      _sum_supply = 0;
667      for (int i = 0; i != _root; ++i) {
668        _sum_supply += _supply[i];
669      }
670      if (_sum_supply > 0) return INFEASIBLE;
671
672      // Check lower and upper bounds
673      LEMON_DEBUG(checkBoundMaps(),
674          "Upper bounds must be greater or equal to the lower bounds");
675
676
677      // Initialize vectors
678      for (int i = 0; i != _res_node_num; ++i) {
679        _pi[i] = 0;
680      }
681      ValueVector excess(_supply);
682
683      // Remove infinite upper bounds and check negative arcs
684      const Value MAX = std::numeric_limits<Value>::max();
685      int last_out;
686      if (_has_lower) {
687        for (int i = 0; i != _root; ++i) {
688          last_out = _first_out[i+1];
689          for (int j = _first_out[i]; j != last_out; ++j) {
690            if (_forward[j]) {
691              Value c = _cost[j] < 0 ? _upper[j] : _lower[j];
692              if (c >= MAX) return UNBOUNDED;
693              excess[i] -= c;
694              excess[_target[j]] += c;
695            }
696          }
697        }
698      } else {
699        for (int i = 0; i != _root; ++i) {
700          last_out = _first_out[i+1];
701          for (int j = _first_out[i]; j != last_out; ++j) {
702            if (_forward[j] && _cost[j] < 0) {
703              Value c = _upper[j];
704              if (c >= MAX) return UNBOUNDED;
705              excess[i] -= c;
706              excess[_target[j]] += c;
707            }
708          }
709        }
710      }
711      Value ex, max_cap = 0;
712      for (int i = 0; i != _res_node_num; ++i) {
713        ex = excess[i];
714        if (ex < 0) max_cap -= ex;
715      }
716      for (int j = 0; j != _res_arc_num; ++j) {
717        if (_upper[j] >= MAX) _upper[j] = max_cap;
718      }
719
720      // Initialize maps for Circulation and remove non-zero lower bounds
721      ConstMap<Arc, Value> low(0);
722      typedef typename Digraph::template ArcMap<Value> ValueArcMap;
723      typedef typename Digraph::template NodeMap<Value> ValueNodeMap;
724      ValueArcMap cap(_graph), flow(_graph);
725      ValueNodeMap sup(_graph);
726      for (NodeIt n(_graph); n != INVALID; ++n) {
727        sup[n] = _supply[_node_id[n]];
728      }
729      if (_has_lower) {
730        for (ArcIt a(_graph); a != INVALID; ++a) {
731          int j = _arc_idf[a];
732          Value c = _lower[j];
733          cap[a] = _upper[j] - c;
734          sup[_graph.source(a)] -= c;
735          sup[_graph.target(a)] += c;
736        }
737      } else {
738        for (ArcIt a(_graph); a != INVALID; ++a) {
739          cap[a] = _upper[_arc_idf[a]];
740        }
741      }
742
743      // Find a feasible flow using Circulation
744      Circulation<Digraph, ConstMap<Arc, Value>, ValueArcMap, ValueNodeMap>
745        circ(_graph, low, cap, sup);
746      if (!circ.flowMap(flow).run()) return INFEASIBLE;
747
748      // Set residual capacities and handle GEQ supply type
749      if (_sum_supply < 0) {
750        for (ArcIt a(_graph); a != INVALID; ++a) {
751          Value fa = flow[a];
752          _res_cap[_arc_idf[a]] = cap[a] - fa;
753          _res_cap[_arc_idb[a]] = fa;
754          sup[_graph.source(a)] -= fa;
755          sup[_graph.target(a)] += fa;
756        }
757        for (NodeIt n(_graph); n != INVALID; ++n) {
758          excess[_node_id[n]] = sup[n];
759        }
760        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
761          int u = _target[a];
762          int ra = _reverse[a];
763          _res_cap[a] = -_sum_supply + 1;
764          _res_cap[ra] = -excess[u];
765          _cost[a] = 0;
766          _cost[ra] = 0;
767        }
768      } else {
769        for (ArcIt a(_graph); a != INVALID; ++a) {
770          Value fa = flow[a];
771          _res_cap[_arc_idf[a]] = cap[a] - fa;
772          _res_cap[_arc_idb[a]] = fa;
773        }
774        for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
775          int ra = _reverse[a];
776          _res_cap[a] = 1;
777          _res_cap[ra] = 0;
778          _cost[a] = 0;
779          _cost[ra] = 0;
780        }
781      }
782
783      return OPTIMAL;
784    }
785
786    // Check if the upper bound is greater than or equal to the lower bound
787    // on each forward arc.
788    bool checkBoundMaps() {
789      for (int j = 0; j != _res_arc_num; ++j) {
790        if (_forward[j] && _upper[j] < _lower[j]) return false;
791      }
792      return true;
793    }
794
795    // Build a StaticDigraph structure containing the current
796    // residual network
797    void buildResidualNetwork() {
798      _arc_vec.clear();
799      _cost_vec.clear();
800      _id_vec.clear();
801      for (int j = 0; j != _res_arc_num; ++j) {
802        if (_res_cap[j] > 0) {
803          _arc_vec.push_back(IntPair(_source[j], _target[j]));
804          _cost_vec.push_back(_cost[j]);
805          _id_vec.push_back(j);
806        }
807      }
808      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
809    }
810
811    // Execute the algorithm and transform the results
812    void start(Method method) {
813      // Execute the algorithm
814      switch (method) {
815        case SIMPLE_CYCLE_CANCELING:
816          startSimpleCycleCanceling();
817          break;
818        case MINIMUM_MEAN_CYCLE_CANCELING:
819          startMinMeanCycleCanceling();
820          break;
821        case CANCEL_AND_TIGHTEN:
822          startCancelAndTighten();
823          break;
824      }
825
826      // Compute node potentials
827      if (method != SIMPLE_CYCLE_CANCELING) {
828        buildResidualNetwork();
829        typename BellmanFord<StaticDigraph, CostArcMap>
830          ::template SetDistMap<CostNodeMap>::Create bf(_sgr, _cost_map);
831        bf.distMap(_pi_map);
832        bf.init(0);
833        bf.start();
834      }
835
836      // Handle non-zero lower bounds
837      if (_has_lower) {
838        int limit = _first_out[_root];
839        for (int j = 0; j != limit; ++j) {
840          if (_forward[j]) _res_cap[_reverse[j]] += _lower[j];
841        }
842      }
843    }
844
845    // Execute the "Simple Cycle Canceling" method
846    void startSimpleCycleCanceling() {
847      // Constants for computing the iteration limits
848      const int BF_FIRST_LIMIT  = 2;
849      const double BF_LIMIT_FACTOR = 1.5;
850
851      typedef StaticVectorMap<StaticDigraph::Arc, Value> FilterMap;
852      typedef FilterArcs<StaticDigraph, FilterMap> ResDigraph;
853      typedef StaticVectorMap<StaticDigraph::Node, StaticDigraph::Arc> PredMap;
854      typedef typename BellmanFord<ResDigraph, CostArcMap>
855        ::template SetDistMap<CostNodeMap>
856        ::template SetPredMap<PredMap>::Create BF;
857
858      // Build the residual network
859      _arc_vec.clear();
860      _cost_vec.clear();
861      for (int j = 0; j != _res_arc_num; ++j) {
862        _arc_vec.push_back(IntPair(_source[j], _target[j]));
863        _cost_vec.push_back(_cost[j]);
864      }
865      _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end());
866
867      FilterMap filter_map(_res_cap);
868      ResDigraph rgr(_sgr, filter_map);
869      std::vector<int> cycle;
870      std::vector<StaticDigraph::Arc> pred(_res_arc_num);
871      PredMap pred_map(pred);
872      BF bf(rgr, _cost_map);
873      bf.distMap(_pi_map).predMap(pred_map);
874
875      int length_bound = BF_FIRST_LIMIT;
876      bool optimal = false;
877      while (!optimal) {
878        bf.init(0);
879        int iter_num = 0;
880        bool cycle_found = false;
881        while (!cycle_found) {
882          // Perform some iterations of the Bellman-Ford algorithm
883          int curr_iter_num = iter_num + length_bound <= _node_num ?
884            length_bound : _node_num - iter_num;
885          iter_num += curr_iter_num;
886          int real_iter_num = curr_iter_num;
887          for (int i = 0; i < curr_iter_num; ++i) {
888            if (bf.processNextWeakRound()) {
889              real_iter_num = i;
890              break;
891            }
892          }
893          if (real_iter_num < curr_iter_num) {
894            // Optimal flow is found
895            optimal = true;
896            break;
897          } else {
898            // Search for node disjoint negative cycles
899            std::vector<int> state(_res_node_num, 0);
900            int id = 0;
901            for (int u = 0; u != _res_node_num; ++u) {
902              if (state[u] != 0) continue;
903              ++id;
904              int v = u;
905              for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ?
906                   -1 : rgr.id(rgr.source(pred[v]))) {
907                state[v] = id;
908              }
909              if (v != -1 && state[v] == id) {
910                // A negative cycle is found
911                cycle_found = true;
912                cycle.clear();
913                StaticDigraph::Arc a = pred[v];
914                Value d, delta = _res_cap[rgr.id(a)];
915                cycle.push_back(rgr.id(a));
916                while (rgr.id(rgr.source(a)) != v) {
917                  a = pred_map[rgr.source(a)];
918                  d = _res_cap[rgr.id(a)];
919                  if (d < delta) delta = d;
920                  cycle.push_back(rgr.id(a));
921                }
922
923                // Augment along the cycle
924                for (int i = 0; i < int(cycle.size()); ++i) {
925                  int j = cycle[i];
926                  _res_cap[j] -= delta;
927                  _res_cap[_reverse[j]] += delta;
928                }
929              }
930            }
931          }
932
933          // Increase iteration limit if no cycle is found
934          if (!cycle_found) {
935            length_bound = static_cast<int>(length_bound * BF_LIMIT_FACTOR);
936          }
937        }
938      }
939    }
940
941    // Execute the "Minimum Mean Cycle Canceling" method
942    void startMinMeanCycleCanceling() {
943      typedef Path<StaticDigraph> SPath;
944      typedef typename SPath::ArcIt SPathArcIt;
945      typedef typename HowardMmc<StaticDigraph, CostArcMap>
946        ::template SetPath<SPath>::Create HwMmc;
947      typedef typename HartmannOrlinMmc<StaticDigraph, CostArcMap>
948        ::template SetPath<SPath>::Create HoMmc;
949
950      const double HW_ITER_LIMIT_FACTOR = 1.0;
951      const int HW_ITER_LIMIT_MIN_VALUE = 5;
952
953      const int hw_iter_limit =
954          std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
955                   HW_ITER_LIMIT_MIN_VALUE);
956
957      SPath cycle;
958      HwMmc hw_mmc(_sgr, _cost_map);
959      hw_mmc.cycle(cycle);
960      buildResidualNetwork();
961      while (true) {
962
963        typename HwMmc::TerminationCause hw_tc =
964            hw_mmc.findCycleMean(hw_iter_limit);
965        if (hw_tc == HwMmc::ITERATION_LIMIT) {
966          // Howard's algorithm reached the iteration limit, start a
967          // strongly polynomial algorithm instead
968          HoMmc ho_mmc(_sgr, _cost_map);
969          ho_mmc.cycle(cycle);
970          // Find a minimum mean cycle (Hartmann-Orlin algorithm)
971          if (!(ho_mmc.findCycleMean() && ho_mmc.cycleCost() < 0)) break;
972          ho_mmc.findCycle();
973        } else {
974          // Find a minimum mean cycle (Howard algorithm)
975          if (!(hw_tc == HwMmc::OPTIMAL && hw_mmc.cycleCost() < 0)) break;
976          hw_mmc.findCycle();
977        }
978
979        // Compute delta value
980        Value delta = INF;
981        for (SPathArcIt a(cycle); a != INVALID; ++a) {
982          Value d = _res_cap[_id_vec[_sgr.id(a)]];
983          if (d < delta) delta = d;
984        }
985
986        // Augment along the cycle
987        for (SPathArcIt a(cycle); a != INVALID; ++a) {
988          int j = _id_vec[_sgr.id(a)];
989          _res_cap[j] -= delta;
990          _res_cap[_reverse[j]] += delta;
991        }
992
993        // Rebuild the residual network
994        buildResidualNetwork();
995      }
996    }
997
998    // Execute the "Cancel-and-Tighten" method
999    void startCancelAndTighten() {
1000      // Constants for the min mean cycle computations
1001      const double LIMIT_FACTOR = 1.0;
1002      const int MIN_LIMIT = 5;
1003      const double HW_ITER_LIMIT_FACTOR = 1.0;
1004      const int HW_ITER_LIMIT_MIN_VALUE = 5;
1005
1006      const int hw_iter_limit =
1007          std::max(static_cast<int>(HW_ITER_LIMIT_FACTOR * _node_num),
1008                   HW_ITER_LIMIT_MIN_VALUE);
1009
1010      // Contruct auxiliary data vectors
1011      DoubleVector pi(_res_node_num, 0.0);
1012      IntVector level(_res_node_num);
1013      BoolVector reached(_res_node_num);
1014      BoolVector processed(_res_node_num);
1015      IntVector pred_node(_res_node_num);
1016      IntVector pred_arc(_res_node_num);
1017      std::vector<int> stack(_res_node_num);
1018      std::vector<int> proc_vector(_res_node_num);
1019
1020      // Initialize epsilon
1021      double epsilon = 0;
1022      for (int a = 0; a != _res_arc_num; ++a) {
1023        if (_res_cap[a] > 0 && -_cost[a] > epsilon)
1024          epsilon = -_cost[a];
1025      }
1026
1027      // Start phases
1028      Tolerance<double> tol;
1029      tol.epsilon(1e-6);
1030      int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num)));
1031      if (limit < MIN_LIMIT) limit = MIN_LIMIT;
1032      int iter = limit;
1033      while (epsilon * _res_node_num >= 1) {
1034        // Find and cancel cycles in the admissible network using DFS
1035        for (int u = 0; u != _res_node_num; ++u) {
1036          reached[u] = false;
1037          processed[u] = false;
1038        }
1039        int stack_head = -1;
1040        int proc_head = -1;
1041        for (int start = 0; start != _res_node_num; ++start) {
1042          if (reached[start]) continue;
1043
1044          // New start node
1045          reached[start] = true;
1046          pred_arc[start] = -1;
1047          pred_node[start] = -1;
1048
1049          // Find the first admissible outgoing arc
1050          double p = pi[start];
1051          int a = _first_out[start];
1052          int last_out = _first_out[start+1];
1053          for (; a != last_out && (_res_cap[a] == 0 ||
1054               !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1055          if (a == last_out) {
1056            processed[start] = true;
1057            proc_vector[++proc_head] = start;
1058            continue;
1059          }
1060          stack[++stack_head] = a;
1061
1062          while (stack_head >= 0) {
1063            int sa = stack[stack_head];
1064            int u = _source[sa];
1065            int v = _target[sa];
1066
1067            if (!reached[v]) {
1068              // A new node is reached
1069              reached[v] = true;
1070              pred_node[v] = u;
1071              pred_arc[v] = sa;
1072              p = pi[v];
1073              a = _first_out[v];
1074              last_out = _first_out[v+1];
1075              for (; a != last_out && (_res_cap[a] == 0 ||
1076                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1077              stack[++stack_head] = a == last_out ? -1 : a;
1078            } else {
1079              if (!processed[v]) {
1080                // A cycle is found
1081                int n, w = u;
1082                Value d, delta = _res_cap[sa];
1083                for (n = u; n != v; n = pred_node[n]) {
1084                  d = _res_cap[pred_arc[n]];
1085                  if (d <= delta) {
1086                    delta = d;
1087                    w = pred_node[n];
1088                  }
1089                }
1090
1091                // Augment along the cycle
1092                _res_cap[sa] -= delta;
1093                _res_cap[_reverse[sa]] += delta;
1094                for (n = u; n != v; n = pred_node[n]) {
1095                  int pa = pred_arc[n];
1096                  _res_cap[pa] -= delta;
1097                  _res_cap[_reverse[pa]] += delta;
1098                }
1099                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {
1100                  --stack_head;
1101                  reached[n] = false;
1102                }
1103                u = w;
1104              }
1105              v = u;
1106
1107              // Find the next admissible outgoing arc
1108              p = pi[v];
1109              a = stack[stack_head] + 1;
1110              last_out = _first_out[v+1];
1111              for (; a != last_out && (_res_cap[a] == 0 ||
1112                   !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1113              stack[stack_head] = a == last_out ? -1 : a;
1114            }
1115
1116            while (stack_head >= 0 && stack[stack_head] == -1) {
1117              processed[v] = true;
1118              proc_vector[++proc_head] = v;
1119              if (--stack_head >= 0) {
1120                // Find the next admissible outgoing arc
1121                v = _source[stack[stack_head]];
1122                p = pi[v];
1123                a = stack[stack_head] + 1;
1124                last_out = _first_out[v+1];
1125                for (; a != last_out && (_res_cap[a] == 0 ||
1126                     !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ;
1127                stack[stack_head] = a == last_out ? -1 : a;
1128              }
1129            }
1130          }
1131        }
1132
1133        // Tighten potentials and epsilon
1134        if (--iter > 0) {
1135          for (int u = 0; u != _res_node_num; ++u) {
1136            level[u] = 0;
1137          }
1138          for (int i = proc_head; i > 0; --i) {
1139            int u = proc_vector[i];
1140            double p = pi[u];
1141            int l = level[u] + 1;
1142            int last_out = _first_out[u+1];
1143            for (int a = _first_out[u]; a != last_out; ++a) {
1144              int v = _target[a];
1145              if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) &&
1146                  l > level[v]) level[v] = l;
1147            }
1148          }
1149
1150          // Modify potentials
1151          double q = std::numeric_limits<double>::max();
1152          for (int u = 0; u != _res_node_num; ++u) {
1153            int lu = level[u];
1154            double p, pu = pi[u];
1155            int last_out = _first_out[u+1];
1156            for (int a = _first_out[u]; a != last_out; ++a) {
1157              if (_res_cap[a] == 0) continue;
1158              int v = _target[a];
1159              int ld = lu - level[v];
1160              if (ld > 0) {
1161                p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1);
1162                if (p < q) q = p;
1163              }
1164            }
1165          }
1166          for (int u = 0; u != _res_node_num; ++u) {
1167            pi[u] -= q * level[u];
1168          }
1169
1170          // Modify epsilon
1171          epsilon = 0;
1172          for (int u = 0; u != _res_node_num; ++u) {
1173            double curr, pu = pi[u];
1174            int last_out = _first_out[u+1];
1175            for (int a = _first_out[u]; a != last_out; ++a) {
1176              if (_res_cap[a] == 0) continue;
1177              curr = _cost[a] + pu - pi[_target[a]];
1178              if (-curr > epsilon) epsilon = -curr;
1179            }
1180          }
1181        } else {
1182          typedef HowardMmc<StaticDigraph, CostArcMap> HwMmc;
1183          typedef HartmannOrlinMmc<StaticDigraph, CostArcMap> HoMmc;
1184          typedef typename BellmanFord<StaticDigraph, CostArcMap>
1185            ::template SetDistMap<CostNodeMap>::Create BF;
1186
1187          // Set epsilon to the minimum cycle mean
1188          Cost cycle_cost = 0;
1189          int cycle_size = 1;
1190          buildResidualNetwork();
1191          HwMmc hw_mmc(_sgr, _cost_map);
1192          if (hw_mmc.findCycleMean(hw_iter_limit) == HwMmc::ITERATION_LIMIT) {
1193            // Howard's algorithm reached the iteration limit, start a
1194            // strongly polynomial algorithm instead
1195            HoMmc ho_mmc(_sgr, _cost_map);
1196            ho_mmc.findCycleMean();
1197            epsilon = -ho_mmc.cycleMean();
1198            cycle_cost = ho_mmc.cycleCost();
1199            cycle_size = ho_mmc.cycleSize();
1200          } else {
1201            // Set epsilon
1202            epsilon = -hw_mmc.cycleMean();
1203            cycle_cost = hw_mmc.cycleCost();
1204            cycle_size = hw_mmc.cycleSize();
1205          }
1206
1207          // Compute feasible potentials for the current epsilon
1208          for (int i = 0; i != int(_cost_vec.size()); ++i) {
1209            _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost;
1210          }
1211          BF bf(_sgr, _cost_map);
1212          bf.distMap(_pi_map);
1213          bf.init(0);
1214          bf.start();
1215          for (int u = 0; u != _res_node_num; ++u) {
1216            pi[u] = static_cast<double>(_pi[u]) / cycle_size;
1217          }
1218
1219          iter = limit;
1220        }
1221      }
1222    }
1223
1224  }; //class CycleCanceling
1225
1226  ///@}
1227
1228} //namespace lemon
1229
1230#endif //LEMON_CYCLE_CANCELING_H
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