RhodeCode

/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_CANCEL_AND_TIGHTEN_H

#define LEMON_CANCEL_AND_TIGHTEN_H

/// \ingroup min_cost_flow

///

/// \file

/// \brief Cancel and Tighten algorithm for finding a minimum cost flow.

#include <vector>

#include <lemon/circulation.h>

#include <lemon/bellman_ford.h>

#include <lemon/howard.h>

#include <lemon/adaptors.h>

#include <lemon/tolerance.h>

#include <lemon/math.h>

#include <lemon/static_graph.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of the Cancel and Tighten algorithm for

  /// finding a minimum cost flow.

  ///

  /// \ref CancelAndTighten implements the Cancel and Tighten algorithm for

  /// finding a minimum cost flow.

  ///

  /// \tparam Digraph The digraph type the algorithm runs on.

  /// \tparam LowerMap The type of the lower bound map.

  /// \tparam CapacityMap The type of the capacity (upper bound) map.

  /// \tparam CostMap The type of the cost (length) map.

  /// \tparam SupplyMap The type of the supply map.

  ///

  /// \warning

  /// - Arc capacities and costs should be \e non-negative \e integers.

  /// - Supply values should be \e signed \e integers.

  /// - The value types of the maps should be convertible to each other.

  /// - \c CostMap::Value must be signed type.

  ///

  /// \author Peter Kovacs

  template < typename Digraph,

             typename LowerMap = typename Digraph::template ArcMap<int>,

             typename CapacityMap = typename Digraph::template ArcMap<int>,

             typename CostMap = typename Digraph::template ArcMap<int>,

             typename SupplyMap = typename Digraph::template NodeMap<int> >

  class CancelAndTighten

  {

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

    typedef typename Digraph::template ArcMap<Capacity> CapacityArcMap;

    typedef typename Digraph::template NodeMap<Supply> SupplyNodeMap;

    typedef ResidualDigraph< const Digraph,

      CapacityArcMap, CapacityArcMap > ResDigraph;

  public:

    /// The type of the flow map.

    typedef typename Digraph::template ArcMap<Capacity> FlowMap;

    /// The type of the potential map.

    typedef typename Digraph::template NodeMap<Cost> PotentialMap;

  private:

    /// \brief Map adaptor class for handling residual arc costs.

    ///

    /// Map adaptor class for handling residual arc costs.

    class ResidualCostMap : public MapBase<typename ResDigraph::Arc, Cost>

    {

      typedef typename ResDigraph::Arc Arc;

    private:

      const CostMap &_cost_map;

    public:

      ///\e

      ResidualCostMap(const CostMap &cost_map) : _cost_map(cost_map) {}

      ///\e

      Cost operator[](const Arc &e) const {

        return ResDigraph::forward(e) ? _cost_map[e] : -_cost_map[e];

      }

    }; //class ResidualCostMap

    /// \brief Map adaptor class for handling reduced arc costs.

    ///

    /// Map adaptor class for handling reduced arc costs.

    class ReducedCostMap : public MapBase<Arc, Cost>

    {

    private:

      const Digraph &_gr;

      const CostMap &_cost_map;

      const PotentialMap &_pot_map;

    public:

      ///\e

      ReducedCostMap( const Digraph &gr,

                      const CostMap &cost_map,

                      const PotentialMap &pot_map ) :

        _gr(gr), _cost_map(cost_map), _pot_map(pot_map) {}

      ///\e

      inline Cost operator[](const Arc &e) const {

        return _cost_map[e] + _pot_map[_gr.source(e)]

                            - _pot_map[_gr.target(e)];

      }

    }; //class ReducedCostMap

    struct BFOperationTraits {

      static double zero() { return 0; }

      static double infinity() {

        return std::numeric_limits<double>::infinity();

      }

      static double plus(const double& left, const double& right) {

        return left + right;

      }

      static bool less(const double& left, const double& right) {

        return left + 1e-6 < right;

      }

    }; // class BFOperationTraits

  private:

    // The digraph the algorithm runs on

    const Digraph &_graph;

    // The original lower bound map

    const LowerMap *_lower;

    // The modified capacity map

    CapacityArcMap _capacity;

    // The original cost map

    const CostMap &_cost;

    // The modified supply map

    SupplyNodeMap _supply;

    bool _valid_supply;

    // Arc map of the current flow

    FlowMap *_flow;

    bool _local_flow;

    // Node map of the current potentials

    PotentialMap *_potential;

    bool _local_potential;

    // The residual digraph

    ResDigraph *_res_graph;

    // The residual cost map

    ResidualCostMap _res_cost;

  public:

    /// \brief General constructor (with lower bounds).

    ///

    /// General constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CancelAndTighten( const Digraph &digraph,

                      const LowerMap &lower,

                      const CapacityMap &capacity,

                      const CostMap &cost,

                      const SupplyMap &supply ) :

      _graph(digraph), _lower(&lower), _capacity(digraph), _cost(cost),

      _supply(digraph), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false),

      _res_graph(NULL), _res_cost(_cost)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[n] = supply[n];

        sum += _supply[n];

      }

      _valid_supply = sum == 0;

      // Remove non-zero lower bounds

      for (ArcIt e(_graph); e != INVALID; ++e) {

        _capacity[e] = capacity[e];

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

    }

/*

    /// \brief General constructor (without lower bounds).

    ///

    /// General constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CancelAndTighten( const Digraph &digraph,

                      const CapacityMap &capacity,

                      const CostMap &cost,

                      const SupplyMap &supply ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost),

      _supply(supply), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false),

      _res_graph(NULL), _res_cost(_cost)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];

      _valid_supply = sum == 0;

    }

    /// \brief Simple constructor (with lower bounds).

    ///

    /// Simple constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value The required amount of flow from node \c s

    /// to node \c t (i.e. the supply of \c s and the demand of \c t).

    CancelAndTighten( const Digraph &digraph,

                      const LowerMap &lower,

                      const CapacityMap &capacity,

                      const CostMap &cost,

                      Node s, Node t,

                      Supply flow_value ) :

      _graph(digraph), _lower(&lower), _capacity(capacity), _cost(cost),

      _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false),

      _res_graph(NULL), _res_cost(_cost)

    {

      // Remove non-zero lower bounds

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      for (ArcIt e(_graph); e != INVALID; ++e) {

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

      _valid_supply = true;

    }

    /// \brief Simple constructor (without lower bounds).

    ///

    /// Simple constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value The required amount of flow from node \c s

    /// to node \c t (i.e. the supply of \c s and the demand of \c t).

    CancelAndTighten( const Digraph &digraph,

                      const CapacityMap &capacity,

                      const CostMap &cost,

                      Node s, Node t,

                      Supply flow_value ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost),

      _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false),

      _res_graph(NULL), _res_cost(_cost)

    {

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      _valid_supply = true;

    }

*/

    /// Destructor.

    ~CancelAndTighten() {

      if (_local_flow) delete _flow;

      if (_local_potential) delete _potential;

      delete _res_graph;

    }

    /// \brief Set the flow map.

    ///

    /// Set the flow map.

    ///

    /// \return \c (*this)

    CancelAndTighten& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow;

        _local_flow = false;

      }

      _flow = ↦

      return *this;

    }

    /// \brief Set the potential map.

    ///

    /// Set the potential map.

    ///

    /// \return \c (*this)

    CancelAndTighten& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential;

        _local_potential = false;

      }

      _potential = ↦

      return *this;

    }

    /// \name Execution control

    /// @{

    /// \brief Run the algorithm.

    ///

    /// Run the algorithm.

    ///

    /// \return \c true if a feasible flow can be found.

    bool run() {

      return init() && start();

    }

    /// @}

    /// \name Query Functions

    /// The result of the algorithm can be obtained using these

    /// functions.\n

    /// \ref lemon::CancelAndTighten::run() "run()" must be called before

    /// using them.

    /// @{

    /// \brief Return a const reference to the arc map storing the

    /// found flow.

    ///

    /// Return a const reference to the arc map storing the found flow.

    ///

    /// \pre \ref run() must be called before using this function.

    const FlowMap& flowMap() const {

      return *_flow;

    }

    /// \brief Return a const reference to the node map storing the

    /// found potentials (the dual solution).

    ///

    /// Return a const reference to the node map storing the found

    /// potentials (the dual solution).

    ///

    /// \pre \ref run() must be called before using this function.

    const PotentialMap& potentialMap() const {

      return *_potential;

    }

    /// \brief Return the flow on the given arc.

    ///

    /// Return the flow on the given arc.

    ///

    /// \pre \ref run() must be called before using this function.

    Capacity flow(const Arc& arc) const {

      return (*_flow)[arc];

    }

    /// \brief Return the potential of the given node.

    ///

    /// Return the potential of the given node.

    ///

    /// \pre \ref run() must be called before using this function.

    Cost potential(const Node& node) const {

      return (*_potential)[node];

    }

    /// \brief Return the total cost of the found flow.

    ///

    /// Return the total cost of the found flow. The complexity of the

    /// function is \f$ O(e) \f$.

    ///

    /// \pre \ref run() must be called before using this function.

    Cost totalCost() const {

      Cost c = 0;

      for (ArcIt e(_graph); e != INVALID; ++e)

        c += (*_flow)[e] * _cost[e];

      return c;

    }

    /// @}

  private:

    /// Initialize the algorithm.

    bool init() {

      if (!_valid_supply) return false;

      // Initialize flow and potential maps

      if (!_flow) {

        _flow = new FlowMap(_graph);

        _local_flow = true;

      }

      if (!_potential) {

        _potential = new PotentialMap(_graph);

        _local_potential = true;

      }

      _res_graph = new ResDigraph(_graph, _capacity, *_flow);

      // Find a feasible flow using Circulation

      Circulation< Digraph, ConstMap<Arc, Capacity>,

                   CapacityArcMap, SupplyMap >

        circulation( _graph, constMap<Arc>(Capacity(0)),

                     _capacity, _supply );

      return circulation.flowMap(*_flow).run();

    }

    bool start() {

      const double LIMIT_FACTOR = 0.01;

      const int MIN_LIMIT = 3;

      typedef typename Digraph::template NodeMap<double> FloatPotentialMap;

      typedef typename Digraph::template NodeMap<int> LevelMap;

      typedef typename Digraph::template NodeMap<bool> BoolNodeMap;

      typedef typename Digraph::template NodeMap<Node> PredNodeMap;

      typedef typename Digraph::template NodeMap<Arc> PredArcMap;

      typedef typename ResDigraph::template ArcMap<double> ResShiftCostMap;

      FloatPotentialMap pi(_graph);

      LevelMap level(_graph);

      BoolNodeMap reached(_graph);

      BoolNodeMap processed(_graph);

      PredNodeMap pred_node(_graph);

      PredArcMap pred_arc(_graph);

      int node_num = countNodes(_graph);

      typedef std::pair<Arc, bool> pair;

      std::vector<pair> stack(node_num);

      std::vector<Node> proc_vector(node_num);

      ResShiftCostMap shift_cost(*_res_graph);

      Tolerance<double> tol;

      tol.epsilon(1e-6);

      Timer t1, t2, t3;

      t1.reset();

      t2.reset();

      t3.reset();

      // Initialize epsilon and the node potentials

      double epsilon = 0;

      for (ArcIt e(_graph); e != INVALID; ++e) {

        if (_capacity[e] - (*_flow)[e] > 0 && _cost[e] < -epsilon)

          epsilon = -_cost[e];

        else if ((*_flow)[e] > 0 && _cost[e] > epsilon)

          epsilon = _cost[e];

      }

      for (NodeIt v(_graph); v != INVALID; ++v) {

        pi[v] = 0;

      }

      // Start phases

      int limit = int(LIMIT_FACTOR * node_num);

      if (limit < MIN_LIMIT) limit = MIN_LIMIT;

      int iter = limit;

      while (epsilon * node_num >= 1) {

        t1.start();

        // Find and cancel cycles in the admissible digraph using DFS

        for (NodeIt n(_graph); n != INVALID; ++n) {

          reached[n] = false;

          processed[n] = false;

        }

        int stack_head = -1;

        int proc_head = -1;

        for (NodeIt start(_graph); start != INVALID; ++start) {

          if (reached[start]) continue;

          // New start node

          reached[start] = true;

          pred_arc[start] = INVALID;

          pred_node[start] = INVALID;

          // Find the first admissible residual outgoing arc

          double p = pi[start];

          Arc e;

          _graph.firstOut(e, start);

          while ( e != INVALID && (_capacity[e] - (*_flow)[e] == 0 ||

                  !tol.negative(_cost[e] + p - pi[_graph.target(e)])) )

            _graph.nextOut(e);

          if (e != INVALID) {

            stack[++stack_head] = pair(e, true);

            goto next_step_1;

          }

          _graph.firstIn(e, start);

          while ( e != INVALID && ((*_flow)[e] == 0 ||

                  !tol.negative(-_cost[e] + p - pi[_graph.source(e)])) )

            _graph.nextIn(e);

          if (e != INVALID) {

            stack[++stack_head] = pair(e, false);

            goto next_step_1;

          }

          processed[start] = true;

          proc_vector[++proc_head] = start;

          continue;

        next_step_1:

          while (stack_head >= 0) {

            Arc se = stack[stack_head].first;

            bool sf = stack[stack_head].second;

            Node u, v;

            if (sf) {

              u = _graph.source(se);

              v = _graph.target(se);

            } else {

              u = _graph.target(se);

              v = _graph.source(se);

            }

            if (!reached[v]) {

              // A new node is reached

              reached[v] = true;

              pred_node[v] = u;

              pred_arc[v] = se;

              // Find the first admissible residual outgoing arc

              double p = pi[v];

              Arc e;

              _graph.firstOut(e, v);

              while ( e != INVALID && (_capacity[e] - (*_flow)[e] == 0 ||

                      !tol.negative(_cost[e] + p - pi[_graph.target(e)])) )

                _graph.nextOut(e);

              if (e != INVALID) {

                stack[++stack_head] = pair(e, true);

                goto next_step_2;

              }

              _graph.firstIn(e, v);

              while ( e != INVALID && ((*_flow)[e] == 0 ||

                      !tol.negative(-_cost[e] + p - pi[_graph.source(e)])) )

                _graph.nextIn(e);

              stack[++stack_head] = pair(e, false);

            next_step_2: ;

            } else {

              if (!processed[v]) {

                // A cycle is found

                Node n, w = u;

                Capacity d, delta = sf ? _capacity[se] - (*_flow)[se] :

                                         (*_flow)[se];

                for (n = u; n != v; n = pred_node[n]) {

                  d = _graph.target(pred_arc[n]) == n ?

                      _capacity[pred_arc[n]] - (*_flow)[pred_arc[n]] :

                      (*_flow)[pred_arc[n]];

                  if (d <= delta) {

                    delta = d;

                    w = pred_node[n];

                  }

                }

/*

                std::cout << "CYCLE FOUND: ";

                if (sf)

                  std::cout << _cost[se] + pi[_graph.source(se)] - pi[_graph.target(se)];

                else

                  std::cout << _graph.id(se) << ":" << -(_cost[se] + pi[_graph.source(se)] - pi[_graph.target(se)]);

                for (n = u; n != v; n = pred_node[n]) {

                  if (_graph.target(pred_arc[n]) == n)

                    std::cout << " " << _cost[pred_arc[n]] + pi[_graph.source(pred_arc[n])] - pi[_graph.target(pred_arc[n])];

                  else

                    std::cout << " " << -(_cost[pred_arc[n]] + pi[_graph.source(pred_arc[n])] - pi[_graph.target(pred_arc[n])]);

                }

                std::cout << "\n";

*/

                // Augment along the cycle

                (*_flow)[se] = sf ? (*_flow)[se] + delta :

                                    (*_flow)[se] - delta;

                for (n = u; n != v; n = pred_node[n]) {

                  if (_graph.target(pred_arc[n]) == n)

                    (*_flow)[pred_arc[n]] += delta;

                  else

                    (*_flow)[pred_arc[n]] -= delta;

                }

                for (n = u; stack_head > 0 && n != w; n = pred_node[n]) {

                  --stack_head;

                  reached[n] = false;

                }

                u = w;

              }

              v = u;

              // Find the next admissible residual outgoing arc

              double p = pi[v];

              Arc e = stack[stack_head].first;

              if (!stack[stack_head].second) {

                _graph.nextIn(e);

                goto in_arc_3;

              }

              _graph.nextOut(e);

              while ( e != INVALID && (_capacity[e] - (*_flow)[e] == 0 ||

                      !tol.negative(_cost[e] + p - pi[_graph.target(e)])) )

                _graph.nextOut(e);

              if (e != INVALID) {

                stack[stack_head] = pair(e, true);

                goto next_step_3;

              }

              _graph.firstIn(e, v);

            in_arc_3:

              while ( e != INVALID && ((*_flow)[e] == 0 ||

                      !tol.negative(-_cost[e] + p - pi[_graph.source(e)])) )

                _graph.nextIn(e);

              stack[stack_head] = pair(e, false);

            next_step_3: ;

            }

            while (stack_head >= 0 && stack[stack_head].first == INVALID) {

              processed[v] = true;

              proc_vector[++proc_head] = v;

              if (--stack_head >= 0) {

                v = stack[stack_head].second ?

                    _graph.source(stack[stack_head].first) :

                    _graph.target(stack[stack_head].first);

                // Find the next admissible residual outgoing arc

                double p = pi[v];

                Arc e = stack[stack_head].first;

                if (!stack[stack_head].second) {

                  _graph.nextIn(e);

                  goto in_arc_4;

                }

                _graph.nextOut(e);

                while ( e != INVALID && (_capacity[e] - (*_flow)[e] == 0 ||

                        !tol.negative(_cost[e] + p - pi[_graph.target(e)])) )

                  _graph.nextOut(e);

                if (e != INVALID) {

                  stack[stack_head] = pair(e, true);

                  goto next_step_4;

                }

                _graph.firstIn(e, v);

              in_arc_4:

                while ( e != INVALID && ((*_flow)[e] == 0 ||

                        !tol.negative(-_cost[e] + p - pi[_graph.source(e)])) )

                  _graph.nextIn(e);

                stack[stack_head] = pair(e, false);

              next_step_4: ;

              }

            }

          }

        }

        t1.stop();

        // Tighten potentials and epsilon

        if (--iter > 0) {

          // Compute levels

          t2.start();

          for (int i = proc_head; i >= 0; --i) {

            Node v = proc_vector[i];

            double p = pi[v];

            int l = 0;

            for (InArcIt e(_graph, v); e != INVALID; ++e) {

              Node u = _graph.source(e);

              if ( _capacity[e] - (*_flow)[e] > 0 &&

                   tol.negative(_cost[e] + pi[u] - p) &&

                   level[u] + 1 > l ) l = level[u] + 1;

            }

            for (OutArcIt e(_graph, v); e != INVALID; ++e) {

              Node u = _graph.target(e);

              if ( (*_flow)[e] > 0 &&

                   tol.negative(-_cost[e] + pi[u] - p) &&

                   level[u] + 1 > l ) l = level[u] + 1;

            }

            level[v] = l;

          }

          // Modify potentials

          double p, q = -1;

          for (ArcIt e(_graph); e != INVALID; ++e) {

            Node u = _graph.source(e);

            Node v = _graph.target(e);

            if (_capacity[e] - (*_flow)[e] > 0 && level[u] - level[v] > 0) {

              p = (_cost[e] + pi[u] - pi[v] + epsilon) /

                  (level[u] - level[v] + 1);

              if (q < 0 || p < q) q = p;

            }

            else if ((*_flow)[e] > 0 && level[v] - level[u] > 0) {

              p = (-_cost[e] - pi[u] + pi[v] + epsilon) /

                  (level[v] - level[u] + 1);

              if (q < 0 || p < q) q = p;

            }

          }

          for (NodeIt v(_graph); v != INVALID; ++v) {

            pi[v] -= q * level[v];

          }

          // Modify epsilon

          epsilon = 0;

          for (ArcIt e(_graph); e != INVALID; ++e) {

            double curr = _cost[e] + pi[_graph.source(e)]

                                   - pi[_graph.target(e)];

            if (_capacity[e] - (*_flow)[e] > 0 && curr < -epsilon)

              epsilon = -curr;

            else if ((*_flow)[e] > 0 && curr > epsilon)

              epsilon = curr;

          }

          t2.stop();

        } else {

          // Set epsilon to the minimum cycle mean

          t3.start();

/**/

          StaticDigraph static_graph;

          typename ResDigraph::template NodeMap<typename StaticDigraph::Node> node_ref(*_res_graph);

          typename ResDigraph::template ArcMap<typename StaticDigraph::Arc> arc_ref(*_res_graph);

          static_graph.build(*_res_graph, node_ref, arc_ref);

          typename StaticDigraph::template NodeMap<double> static_pi(static_graph);

          typename StaticDigraph::template ArcMap<double> static_cost(static_graph);

          for (typename ResDigraph::ArcIt e(*_res_graph); e != INVALID; ++e)

            static_cost[arc_ref[e]] = _res_cost[e];

          Howard<StaticDigraph, typename StaticDigraph::template ArcMap<double> >

            mmc(static_graph, static_cost);

          mmc.findMinMean();

          epsilon = -mmc.cycleMean();

/**/

/*

          Howard<ResDigraph, ResidualCostMap> mmc(*_res_graph, _res_cost);

          mmc.findMinMean();

          epsilon = -mmc.cycleMean();

*/

          // Compute feasible potentials for the current epsilon

          for (typename StaticDigraph::ArcIt e(static_graph); e != INVALID; ++e)

            static_cost[e] += epsilon;

          typename BellmanFord<StaticDigraph, typename StaticDigraph::template ArcMap<double> >::

            template SetDistMap<typename StaticDigraph::template NodeMap<double> >::

            template SetOperationTraits<BFOperationTraits>::Create

              bf(static_graph, static_cost);

          bf.distMap(static_pi).init(0);

          bf.start();

          for (NodeIt n(_graph); n != INVALID; ++n)

            pi[n] = static_pi[node_ref[n]];

/*

          for (typename ResDigraph::ArcIt e(*_res_graph); e != INVALID; ++e)

            shift_cost[e] = _res_cost[e] + epsilon;

          typename BellmanFord<ResDigraph, ResShiftCostMap>::

            template SetDistMap<FloatPotentialMap>::

            template SetOperationTraits<BFOperationTraits>::Create

              bf(*_res_graph, shift_cost);

          bf.distMap(pi).init(0);

          bf.start();

*/

          iter = limit;

          t3.stop();

        }

      }

//      std::cout << t1.realTime() << " " << t2.realTime() << " " << t3.realTime() << "\n";

      // Handle non-zero lower bounds

      if (_lower) {

        for (ArcIt e(_graph); e != INVALID; ++e)

          (*_flow)[e] += (*_lower)[e];

      }

      return true;

    }

  }; //class CancelAndTighten

  ///@}

} //namespace lemon

#endif //LEMON_CANCEL_AND_TIGHTEN_H

/* -*- C++ -*-

 *

 * This file is a part of LEMON, a generic C++ optimization library

 *

 * Copyright (C) 2003-2008

 * Egervary Jeno Kombinatorikus Optimalizalasi Kutatocsoport

 * (Egervary Research Group on Combinatorial Optimization, EGRES).

 *

 * Permission to use, modify and distribute this software is granted

 * provided that this copyright notice appears in all copies. For

 * precise terms see the accompanying LICENSE file.

 *

 * This software is provided "AS IS" with no warranty of any kind,

 * express or implied, and with no claim as to its suitability for any

 * purpose.

 *

 */

#ifndef LEMON_CYCLE_CANCELING_H

#define LEMON_CYCLE_CANCELING_H

/// \ingroup min_cost_flow

///

/// \file

/// \brief Cycle-canceling algorithm for finding a minimum cost flow.

#include <vector>

#include <lemon/adaptors.h>

#include <lemon/path.h>

#include <lemon/circulation.h>

#include <lemon/bellman_ford.h>

#include <lemon/howard.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of a cycle-canceling algorithm for

  /// finding a minimum cost flow.

  ///

  /// \ref CycleCanceling implements a cycle-canceling algorithm for

  /// finding a minimum cost flow.

  ///

  /// \tparam Digraph The digraph type the algorithm runs on.

  /// \tparam LowerMap The type of the lower bound map.

  /// \tparam CapacityMap The type of the capacity (upper bound) map.

  /// \tparam CostMap The type of the cost (length) map.

  /// \tparam SupplyMap The type of the supply map.

  ///

  /// \warning

  /// - Arc capacities and costs should be \e non-negative \e integers.

  /// - Supply values should be \e signed \e integers.

  /// - The value types of the maps should be convertible to each other.

  /// - \c CostMap::Value must be signed type.

  ///

  /// \note By default the \ref BellmanFord "Bellman-Ford" algorithm is

  /// used for negative cycle detection with limited iteration number.

  /// However \ref CycleCanceling also provides the "Minimum Mean

  /// Cycle-Canceling" algorithm, which is \e strongly \e polynomial,

  /// but rather slower in practice.

  /// To use this version of the algorithm, call \ref run() with \c true

  /// parameter.

  ///

  /// \author Peter Kovacs

  template < typename Digraph,

             typename LowerMap = typename Digraph::template ArcMap<int>,

             typename CapacityMap = typename Digraph::template ArcMap<int>,

             typename CostMap = typename Digraph::template ArcMap<int>,

             typename SupplyMap = typename Digraph::template NodeMap<int> >

  class CycleCanceling

  {

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    typedef typename CapacityMap::Value Capacity;

    typedef typename CostMap::Value Cost;

    typedef typename SupplyMap::Value Supply;

    typedef typename Digraph::template ArcMap<Capacity> CapacityArcMap;

    typedef typename Digraph::template NodeMap<Supply> SupplyNodeMap;

    typedef ResidualDigraph< const Digraph,

      CapacityArcMap, CapacityArcMap > ResDigraph;

    typedef typename ResDigraph::Node ResNode;

    typedef typename ResDigraph::NodeIt ResNodeIt;

    typedef typename ResDigraph::Arc ResArc;

    typedef typename ResDigraph::ArcIt ResArcIt;

  public:

    /// The type of the flow map.

    typedef typename Digraph::template ArcMap<Capacity> FlowMap;

    /// The type of the potential map.

    typedef typename Digraph::template NodeMap<Cost> PotentialMap;

  private:

    /// \brief Map adaptor class for handling residual arc costs.

    ///

    /// Map adaptor class for handling residual arc costs.

    class ResidualCostMap : public MapBase<ResArc, Cost>

    {

    private:

      const CostMap &_cost_map;

    public:

      ///\e

      ResidualCostMap(const CostMap &cost_map) : _cost_map(cost_map) {}

      ///\e

      Cost operator[](const ResArc &e) const {

        return ResDigraph::forward(e) ? _cost_map[e] : -_cost_map[e];

      }

    }; //class ResidualCostMap

  private:

    // The maximum number of iterations for the first execution of the

    // Bellman-Ford algorithm. It should be at least 2.

    static const int BF_FIRST_LIMIT  = 2;

    // The iteration limit for the Bellman-Ford algorithm is multiplied

    // by BF_LIMIT_FACTOR/100 in every round.

    static const int BF_LIMIT_FACTOR = 150;

  private:

    // The digraph the algorithm runs on

    const Digraph &_graph;

    // The original lower bound map

    const LowerMap *_lower;

    // The modified capacity map

    CapacityArcMap _capacity;

    // The original cost map

    const CostMap &_cost;

    // The modified supply map

    SupplyNodeMap _supply;

    bool _valid_supply;

    // Arc map of the current flow

    FlowMap *_flow;

    bool _local_flow;

    // Node map of the current potentials

    PotentialMap *_potential;

    bool _local_potential;

    // The residual digraph

    ResDigraph *_res_graph;

    // The residual cost map

    ResidualCostMap _res_cost;

  public:

    /// \brief General constructor (with lower bounds).

    ///

    /// General constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CycleCanceling( const Digraph &digraph,

                    const LowerMap &lower,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    const SupplyMap &supply ) :

      _graph(digraph), _lower(&lower), _capacity(digraph), _cost(cost),

      _supply(digraph), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false),

      _res_graph(NULL), _res_cost(_cost)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) {

        _supply[n] = supply[n];

        sum += _supply[n];

      }

      _valid_supply = sum == 0;

      // Remove non-zero lower bounds

      for (ArcIt e(_graph); e != INVALID; ++e) {

        _capacity[e] = capacity[e];

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

    }

/*

    /// \brief General constructor (without lower bounds).

    ///

    /// General constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param supply The supply values of the nodes (signed).

    CycleCanceling( const Digraph &digraph,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    const SupplyMap &supply ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost),

      _supply(supply), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_graph(NULL),

      _res_cost(_cost)

    {

      // Check the sum of supply values

      Supply sum = 0;

      for (NodeIt n(_graph); n != INVALID; ++n) sum += _supply[n];

      _valid_supply = sum == 0;

    }

    /// \brief Simple constructor (with lower bounds).

    ///

    /// Simple constructor (with lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param lower The lower bounds of the arcs.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value The required amount of flow from node \c s

    /// to node \c t (i.e. the supply of \c s and the demand of \c t).

    CycleCanceling( const Digraph &digraph,

                    const LowerMap &lower,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    Node s, Node t,

                    Supply flow_value ) :

      _graph(digraph), _lower(&lower), _capacity(capacity), _cost(cost),

      _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_graph(NULL),

      _res_cost(_cost)

    {

      // Remove non-zero lower bounds

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      for (ArcIt e(_graph); e != INVALID; ++e) {

        if (lower[e] != 0) {

          _capacity[e] -= lower[e];

          _supply[_graph.source(e)] -= lower[e];

          _supply[_graph.target(e)] += lower[e];

        }

      }

      _valid_supply = true;

    }

    /// \brief Simple constructor (without lower bounds).

    ///

    /// Simple constructor (without lower bounds).

    ///

    /// \param digraph The digraph the algorithm runs on.

    /// \param capacity The capacities (upper bounds) of the arcs.

    /// \param cost The cost (length) values of the arcs.

    /// \param s The source node.

    /// \param t The target node.

    /// \param flow_value The required amount of flow from node \c s

    /// to node \c t (i.e. the supply of \c s and the demand of \c t).

    CycleCanceling( const Digraph &digraph,

                    const CapacityMap &capacity,

                    const CostMap &cost,

                    Node s, Node t,

                    Supply flow_value ) :

      _graph(digraph), _lower(NULL), _capacity(capacity), _cost(cost),

      _supply(digraph, 0), _flow(NULL), _local_flow(false),

      _potential(NULL), _local_potential(false), _res_graph(NULL),

      _res_cost(_cost)

    {

      _supply[s] =  flow_value;

      _supply[t] = -flow_value;

      _valid_supply = true;

    }

*/

    /// Destructor.

    ~CycleCanceling() {

      if (_local_flow) delete _flow;

      if (_local_potential) delete _potential;

      delete _res_graph;

    }

    /// \brief Set the flow map.

    ///

    /// Set the flow map.

    ///

    /// \return \c (*this)

    CycleCanceling& flowMap(FlowMap &map) {

      if (_local_flow) {

        delete _flow;

        _local_flow = false;

      }

      _flow = ↦

      return *this;

    }

    /// \brief Set the potential map.

    ///

    /// Set the potential map.

    ///

    /// \return \c (*this)

    CycleCanceling& potentialMap(PotentialMap &map) {

      if (_local_potential) {

        delete _potential;

        _local_potential = false;

      }

      _potential = ↦

      return *this;

    }

    /// \name Execution control

    /// @{