RhodeCode

/* -*- mode: C++; indent-tabs-mode: nil; -*-

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_CIRCULATION_H

#define LEMON_CIRCULATION_H

#include <lemon/tolerance.h>

#include <lemon/elevator.h>

#include <limits>

///\ingroup max_flow

///\file

///\brief Push-relabel algorithm for finding a feasible circulation.

///

namespace lemon {

  /// \brief Default traits class of Circulation class.

  ///

  /// Default traits class of Circulation class.

  ///

  /// \tparam GR Type of the digraph the algorithm runs on.

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

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

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

  template <typename GR, typename LM,

            typename UM, typename SM>

  struct CirculationDefaultTraits {

    /// \brief The type of the digraph the algorithm runs on.

    typedef GR Digraph;

    /// \brief The type of the lower bound map.

    ///

    /// The type of the map that stores the lower bounds on the arcs.

    /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.

    typedef LM LowerMap;

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

    ///

    /// The type of the map that stores the upper bounds (capacities)

    /// on the arcs.

    /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.

    typedef UM UpperMap;

    /// \brief The type of supply map.

    ///

    /// The type of the map that stores the signed supply values of the 

    /// nodes. 

    /// It must conform to the \ref concepts::ReadMap "ReadMap" concept.

    typedef SM SupplyMap;

    /// \brief The type of the map that stores the flow values.

    ///

    /// The type of the map that stores the flow values.

    /// It must conform to the \ref concepts::ReadWriteMap "ReadWriteMap"

    /// concept.

    /// \brief Instantiates a FlowMap.

    ///

    /// This function instantiates a \ref FlowMap.

    /// \param digraph The digraph for which we would like to define

    /// the flow map.

    static FlowMap* createFlowMap(const Digraph& digraph) {

      return new FlowMap(digraph);

    }

    /// \brief The elevator type used by the algorithm.

    ///

    /// The elevator type used by the algorithm.

    ///

    /// \sa Elevator

    /// \sa LinkedElevator

    typedef lemon::Elevator<Digraph, typename Digraph::Node> Elevator;

    /// \brief Instantiates an Elevator.

    ///

    /// This function instantiates an \ref Elevator.

    /// \param digraph The digraph for which we would like to define

    /// the elevator.

    /// \param max_level The maximum level of the elevator.

    static Elevator* createElevator(const Digraph& digraph, int max_level) {

      return new Elevator(digraph, max_level);

    }

    /// \brief The tolerance used by the algorithm

    ///

    /// The tolerance used by the algorithm to handle inexact computation.

  };

  /**

     \brief Push-relabel algorithm for the network circulation problem.

     \ingroup max_flow

     This class implements a push-relabel algorithm for the \e network

     \e circulation problem.

     It is to find a feasible circulation when lower and upper bounds

     are given for the flow values on the arcs and lower bounds are

     given for the difference between the outgoing and incoming flow

     at the nodes.

     The exact formulation of this problem is the following.

     Let \f$G=(V,A)\f$ be a digraph, \f$lower: A\rightarrow\mathbf{R}\f$

     \f$upper: A\rightarrow\mathbf{R}\cup\{\infty\}\f$ denote the lower and

     upper bounds on the arcs, for which \f$lower(uv) \leq upper(uv)\f$

     holds for all \f$uv\in A\f$, and \f$sup: V\rightarrow\mathbf{R}\f$

     denotes the signed supply values of the nodes.

     If \f$sup(u)>0\f$, then \f$u\f$ is a supply node with \f$sup(u)\f$

     supply, if \f$sup(u)<0\f$, then \f$u\f$ is a demand node with

     \f$-sup(u)\f$ demand.

     A feasible circulation is an \f$f: A\rightarrow\mathbf{R}\f$

     solution of the following problem.

     \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu)

     \geq sup(u) \quad \forall u\in V, \f]

     \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A. \f]

     The sum of the supply values, i.e. \f$\sum_{u\in V} sup(u)\f$ must be

     zero or negative in order to have a feasible solution (since the sum

     of the expressions on the left-hand side of the inequalities is zero).

     It means that the total demand must be greater or equal to the total

     supply and all the supplies have to be carried out from the supply nodes,

     but there could be demands that are not satisfied.

     If \f$\sum_{u\in V} sup(u)\f$ is zero, then all the supply/demand

     constraints have to be satisfied with equality, i.e. all demands

     have to be satisfied and all supplies have to be used.

     If you need the opposite inequalities in the supply/demand constraints

     (i.e. the total demand is less than the total supply and all the demands

     have to be satisfied while there could be supplies that are not used),

     then you could easily transform the problem to the above form by reversing

     the direction of the arcs and taking the negative of the supply values

     (e.g. using \ref ReverseDigraph and \ref NegMap adaptors).

     This algorithm either calculates a feasible circulation, or provides

     a \ref barrier() "barrier", which prooves that a feasible soultion

     cannot exist.

     Note that this algorithm also provides a feasible solution for the

     \ref min_cost_flow "minimum cost flow problem".

     \tparam GR The type of the digraph the algorithm runs on.

     \tparam LM The type of the lower bound map. The default

     map type is \ref concepts::Digraph::ArcMap "GR::ArcMap<int>".

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

     The default map type is \c LM.

     \tparam SM The type of the supply map. The default map type is

     \ref concepts::Digraph::NodeMap "GR::NodeMap<UM::Value>".

  */

#ifdef DOXYGEN

template< typename GR,

          typename LM,

          typename UM,

          typename SM,

          typename TR >

#else

template< typename GR,

          typename LM = typename GR::template ArcMap<int>,

          typename UM = LM,

          typename SM = typename GR::template NodeMap<typename UM::Value>,

          typename TR = CirculationDefaultTraits<GR, LM, UM, SM> >

#endif

  class Circulation {

  public:

    ///The \ref CirculationDefaultTraits "traits class" of the algorithm.

    typedef TR Traits;

    ///The type of the digraph the algorithm runs on.

    typedef typename Traits::Digraph Digraph;

    ///The type of the lower bound map.

    typedef typename Traits::LowerMap LowerMap;

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

    typedef typename Traits::UpperMap UpperMap;

    ///The type of the supply map.

    typedef typename Traits::SupplyMap SupplyMap;

    ///The type of the flow map.

    typedef typename Traits::FlowMap FlowMap;

    ///The type of the elevator.

    typedef typename Traits::Elevator Elevator;

    ///The type of the tolerance.

    typedef typename Traits::Tolerance Tolerance;

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(Digraph);

    const Digraph &_g;

    int _node_num;

    const LowerMap *_lo;

    const UpperMap *_up;

    const SupplyMap *_supply;

    FlowMap *_flow;

    bool _local_flow;

    Elevator* _level;

    bool _local_level;

    ExcessMap* _excess;

    Tolerance _tol;

    int _el;

  public:

    typedef Circulation Create;

    ///\name Named Template Parameters

    ///@{

    template <typename T>

    struct SetFlowMapTraits : public Traits {

      typedef T FlowMap;

      static FlowMap *createFlowMap(const Digraph&) {

        LEMON_ASSERT(false, "FlowMap is not initialized");

        return 0; // ignore warnings

      }

    };

    /// \brief \ref named-templ-param "Named parameter" for setting

    /// FlowMap type

    ///

    /// \ref named-templ-param "Named parameter" for setting FlowMap

    /// type.

    template <typename T>

    struct SetFlowMap

      : public Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                           SetFlowMapTraits<T> > {

      typedef Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                          SetFlowMapTraits<T> > Create;

    };

    template <typename T>

    struct SetElevatorTraits : public Traits {

      typedef T Elevator;

      static Elevator *createElevator(const Digraph&, int) {

        LEMON_ASSERT(false, "Elevator is not initialized");

        return 0; // ignore warnings

      }

    };

    /// \brief \ref named-templ-param "Named parameter" for setting

    /// Elevator type

    ///

    /// \ref named-templ-param "Named parameter" for setting Elevator

    /// type. If this named parameter is used, then an external

    /// elevator object must be passed to the algorithm using the

    /// \ref elevator(Elevator&) "elevator()" function before calling

    /// \ref run() or \ref init().

    /// \sa SetStandardElevator

    template <typename T>

    struct SetElevator

      : public Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                           SetElevatorTraits<T> > {

      typedef Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                          SetElevatorTraits<T> > Create;

    };

    template <typename T>

    struct SetStandardElevatorTraits : public Traits {

      typedef T Elevator;

      static Elevator *createElevator(const Digraph& digraph, int max_level) {

        return new Elevator(digraph, max_level);

      }

    };

    /// \brief \ref named-templ-param "Named parameter" for setting

    /// Elevator type with automatic allocation

    ///

    /// \ref named-templ-param "Named parameter" for setting Elevator

    /// type with automatic allocation.

    /// The Elevator should have standard constructor interface to be

    /// able to automatically created by the algorithm (i.e. the

    /// digraph and the maximum level should be passed to it).

    /// However an external elevator object could also be passed to the

    /// algorithm with the \ref elevator(Elevator&) "elevator()" function

    /// before calling \ref run() or \ref init().

    /// \sa SetElevator

    template <typename T>

    struct SetStandardElevator

      : public Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                       SetStandardElevatorTraits<T> > {

      typedef Circulation<Digraph, LowerMap, UpperMap, SupplyMap,

                      SetStandardElevatorTraits<T> > Create;

    };

    /// @}

  protected:

    Circulation() {}

  public:

        _local_level = false;

      }

      _level = &elevator;

      return *this;

    }

    /// \brief Returns a const reference to the elevator.

    ///

    /// Returns a const reference to the elevator.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    const Elevator& elevator() const {

      return *_level;

    }

    /// \brief Sets the tolerance used by algorithm.

    ///

    /// Sets the tolerance used by algorithm.

    Circulation& tolerance(const Tolerance& tolerance) const {

      _tol = tolerance;

      return *this;

    }

    /// \brief Returns a const reference to the tolerance.

    ///

    /// Returns a const reference to the tolerance.

    const Tolerance& tolerance() const {

      return tolerance;

    }

    /// \name Execution Control

    /// The simplest way to execute the algorithm is to call \ref run().\n

    /// If you need more control on the initial solution or the execution,

    /// first you have to call one of the \ref init() functions, then

    /// the \ref start() function.

    ///@{

    /// Initializes the internal data structures.

    /// Initializes the internal data structures and sets all flow values

    /// to the lower bound.

    void init()

    {

      LEMON_DEBUG(checkBoundMaps(),

        "Upper bounds must be greater or equal to the lower bounds");

      createStructures();

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

        (*_excess)[n] = (*_supply)[n];

      }

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

        _flow->set(e, (*_lo)[e]);

        (*_excess)[_g.target(e)] += (*_flow)[e];

        (*_excess)[_g.source(e)] -= (*_flow)[e];

      }

      // global relabeling tested, but in general case it provides

      // worse performance for random digraphs

      _level->initStart();

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

        _level->initAddItem(n);

      _level->initFinish();

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

        if(_tol.positive((*_excess)[n]))

          _level->activate(n);

    }

    /// Initializes the internal data structures using a greedy approach.

    /// Initializes the internal data structures using a greedy approach

    /// to construct the initial solution.

    void greedyInit()

    {

      LEMON_DEBUG(checkBoundMaps(),

        "Upper bounds must be greater or equal to the lower bounds");

      createStructures();

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

        (*_excess)[n] = (*_supply)[n];

      }

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

        if (!_tol.less(-(*_excess)[_g.target(e)], (*_up)[e])) {

          _flow->set(e, (*_up)[e]);

          (*_excess)[_g.target(e)] += (*_up)[e];

          (*_excess)[_g.source(e)] -= (*_up)[e];

        } else if (_tol.less(-(*_excess)[_g.target(e)], (*_lo)[e])) {

          _flow->set(e, (*_lo)[e]);

          (*_excess)[_g.target(e)] += (*_lo)[e];

          (*_excess)[_g.source(e)] -= (*_lo)[e];

        } else {

          _flow->set(e, fc);

          (*_excess)[_g.target(e)] = 0;

          (*_excess)[_g.source(e)] -= fc;

        }

      }

      _level->initStart();

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

        _level->initAddItem(n);

      _level->initFinish();

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

        if(_tol.positive((*_excess)[n]))

          _level->activate(n);

    }

    ///Executes the algorithm

    ///This function executes the algorithm.

    ///

    ///\return \c true if a feasible circulation is found.

    ///

    ///\sa barrier()

    ///\sa barrierMap()

    bool start()

    {

      Node act;

      Node bact=INVALID;

      Node last_activated=INVALID;

      while((act=_level->highestActive())!=INVALID) {

        int actlevel=(*_level)[act];

        int mlevel=_node_num;

        for(OutArcIt e(_g,act);e!=INVALID; ++e) {

          Node v = _g.target(e);

          if(!_tol.positive(fc)) continue;

          if((*_level)[v]<actlevel) {

            if(!_tol.less(fc, exc)) {

              _flow->set(e, (*_flow)[e] + exc);

              (*_excess)[v] += exc;

              if(!_level->active(v) && _tol.positive((*_excess)[v]))

                _level->activate(v);

              (*_excess)[act] = 0;

              _level->deactivate(act);

              goto next_l;

            }

            else {

              _flow->set(e, (*_up)[e]);

              (*_excess)[v] += fc;

              if(!_level->active(v) && _tol.positive((*_excess)[v]))

                _level->activate(v);

              exc-=fc;

            }

          }

          else if((*_level)[v]<mlevel) mlevel=(*_level)[v];

        }

        for(InArcIt e(_g,act);e!=INVALID; ++e) {

          Node v = _g.source(e);

          if(!_tol.positive(fc)) continue;

          if((*_level)[v]<actlevel) {

            if(!_tol.less(fc, exc)) {

              _flow->set(e, (*_flow)[e] - exc);

              (*_excess)[v] += exc;

              if(!_level->active(v) && _tol.positive((*_excess)[v]))

                _level->activate(v);

              (*_excess)[act] = 0;

              _level->deactivate(act);

              goto next_l;

            }

            else {

              _flow->set(e, (*_lo)[e]);

              (*_excess)[v] += fc;

              if(!_level->active(v) && _tol.positive((*_excess)[v]))

                _level->activate(v);

              exc-=fc;

            }

          }

          else if((*_level)[v]<mlevel) mlevel=(*_level)[v];

        }

        (*_excess)[act] = exc;

        if(!_tol.positive(exc)) _level->deactivate(act);

        else if(mlevel==_node_num) {

          _level->liftHighestActiveToTop();

          _el = _node_num;

          return false;

        }

        else {

          _level->liftHighestActive(mlevel+1);

          if(_level->onLevel(actlevel)==0) {

            _el = actlevel;

            return false;

          }

        }

      next_l:

        ;

      }

      return true;

    }

    /// Runs the algorithm.

    /// This function runs the algorithm.

    ///

    /// \return \c true if a feasible circulation is found.

    ///

    /// \note Apart from the return value, c.run() is just a shortcut of

    /// the following code.

    /// \code

    ///   c.greedyInit();

    ///   c.start();

    /// \endcode

    bool run() {

      greedyInit();

      return start();

    }

    /// @}

    /// \name Query Functions

    /// The results of the circulation algorithm can be obtained using

    /// these functions.\n

    /// Either \ref run() or \ref start() should be called before

    /// using them.

    ///@{

    ///

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

      return (*_flow)[arc];

    }

    /// \brief Returns a const reference to the flow map.

    ///

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

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    const FlowMap& flowMap() const {

      return *_flow;

    }

    /**

       \brief Returns \c true if the given node is in a barrier.

       Barrier is a set \e B of nodes for which

       \f[ \sum_{uv\in A: u\in B} upper(uv) -

           \sum_{uv\in A: v\in B} lower(uv) < \sum_{v\in B} sup(v) \f]

       holds. The existence of a set with this property prooves that a

       feasible circualtion cannot exist.

       This function returns \c true if the given node is in the found

       barrier. If a feasible circulation is found, the function

       gives back \c false for every node.

       \pre Either \ref run() or \ref init() must be called before

       using this function.

       \sa barrierMap()

       \sa checkBarrier()

    */

    bool barrier(const Node& node) const

    {

      return (*_level)[node] >= _el;

    }

    /// \brief Gives back a barrier.

    ///

    /// This function sets \c bar to the characteristic vector of the

    /// found barrier. \c bar should be a \ref concepts::WriteMap "writable"

    /// node map with \c bool (or convertible) value type.

    ///

    /// If a feasible circulation is found, the function gives back an

    /// empty set, so \c bar[v] will be \c false for all nodes \c v.

    ///

    /// \note This function calls \ref barrier() for each node,

    /// so it runs in O(n) time.

    ///

    /// \pre Either \ref run() or \ref init() must be called before

    /// using this function.

    ///

    /// \sa barrier()

    /// \sa checkBarrier()

    template<class BarrierMap>

    void barrierMap(BarrierMap &bar) const

    {

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

        bar.set(n, (*_level)[n] >= _el);

    }

    /// @}

    /// \name Checker Functions

    /// The feasibility of the results can be checked using

    /// these functions.\n

    /// Either \ref run() or \ref start() should be called before

    /// using them.

    ///@{

    ///Check if the found flow is a feasible circulation

    ///Check if the found flow is a feasible circulation,

    ///

    bool checkFlow() const {

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

        if((*_flow)[e]<(*_lo)[e]||(*_flow)[e]>(*_up)[e]) return false;

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

        {

          for(InArcIt e(_g,n);e!=INVALID;++e) dif-=(*_flow)[e];

          for(OutArcIt e(_g,n);e!=INVALID;++e) dif+=(*_flow)[e];

          if(_tol.negative(dif)) return false;

        }

      return true;

    }

    ///Check whether or not the last execution provides a barrier

    ///Check whether or not the last execution provides a barrier.

    ///\sa barrier()

    ///\sa barrierMap()

    bool checkBarrier() const

    {

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

        if(barrier(n))

          delta-=(*_supply)[n];

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

        {

          Node s=_g.source(e);

          Node t=_g.target(e);

          if(barrier(s)&&!barrier(t)) {

            if (_tol.less(inf_cap - (*_up)[e], delta)) return false;

            delta+=(*_up)[e];

          }

          else if(barrier(t)&&!barrier(s)) delta-=(*_lo)[e];

        }

      return _tol.negative(delta);

    }

    /// @}

  };

}

#endif

/* -*- mode: C++; indent-tabs-mode: nil; -*-

 *

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

 *

 * Copyright (C) 2003-2009

 * 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_NETWORK_SIMPLEX_H

#define LEMON_NETWORK_SIMPLEX_H

/// \ingroup min_cost_flow

///

/// \file

/// \brief Network Simplex algorithm for finding a minimum cost flow.

#include <vector>

#include <limits>

#include <algorithm>

#include <lemon/core.h>

#include <lemon/math.h>

namespace lemon {

  /// \addtogroup min_cost_flow

  /// @{

  /// \brief Implementation of the primal Network Simplex algorithm

  /// for finding a \ref min_cost_flow "minimum cost flow".

  ///

  /// \ref NetworkSimplex implements the primal Network Simplex algorithm

  /// for finding a \ref min_cost_flow "minimum cost flow".

  /// This algorithm is a specialized version of the linear programming

  /// simplex method directly for the minimum cost flow problem.

  /// It is one of the most efficient solution methods.

  ///

  /// In general this class is the fastest implementation available

  /// in LEMON for the minimum cost flow problem.

  /// Moreover it supports both directions of the supply/demand inequality

  /// constraints. For more information see \ref SupplyType.

  ///

  /// Most of the parameters of the problem (except for the digraph)

  /// can be given using separate functions, and the algorithm can be

  /// executed using the \ref run() function. If some parameters are not

  /// specified, then default values will be used.

  ///

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

  /// and supply values in the algorithm. By default it is \c int.

  /// \tparam C The value type used for costs and potentials in the

  ///

  /// \warning Both value types must be signed and all input data must

  /// be integer.

  ///

  /// \note %NetworkSimplex provides five different pivot rule

  /// implementations, from which the most efficient one is used

  /// by default. For more information see \ref PivotRule.

  class NetworkSimplex

  {

  public:

    /// The flow type of the algorithm

    /// The cost type of the algorithm

    typedef C Cost;

#ifdef DOXYGEN

    /// The type of the flow map

    /// The type of the potential map

    typedef GR::NodeMap<Cost> PotentialMap;

#else

    /// The type of the flow map

    /// The type of the potential map

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

#endif

  public:

    /// \brief Problem type constants for the \c run() function.

    ///

    /// Enum type containing the problem type constants that can be

    /// returned by the \ref run() function of the algorithm.

    enum ProblemType {

      /// The problem has no feasible solution (flow).

      INFEASIBLE,

      /// The problem has optimal solution (i.e. it is feasible and

      /// bounded), and the algorithm has found optimal flow and node

      /// potentials (primal and dual solutions).

      OPTIMAL,

      /// The objective function of the problem is unbounded, i.e.

      /// there is a directed cycle having negative total cost and

      /// infinite upper bound.

      UNBOUNDED

    };

    /// \brief Constants for selecting the type of the supply constraints.

    ///

    /// Enum type containing constants for selecting the supply type,

    /// i.e. the direction of the inequalities in the supply/demand

    /// constraints of the \ref min_cost_flow "minimum cost flow problem".

    ///

    /// The default supply type is \c GEQ, since this form is supported

    /// by other minimum cost flow algorithms and the \ref Circulation

    /// algorithm, as well.

    /// The \c LEQ problem type can be selected using the \ref supplyType()

    /// function.

    ///

    /// Note that the equality form is a special case of both supply types.

    enum SupplyType {

      /// This option means that there are <em>"greater or equal"</em>

      /// supply/demand constraints in the definition, i.e. the exact

      /// formulation of the problem is the following.

      /**

          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]

          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \geq

              sup(u) \quad \forall u\in V \f]

          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]

      */

      /// It means that the total demand must be greater or equal to the 

      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or

      /// negative) and all the supplies have to be carried out from 

      /// the supply nodes, but there could be demands that are not 

      /// satisfied.

      GEQ,

      /// It is just an alias for the \c GEQ option.

      CARRY_SUPPLIES = GEQ,

      /// This option means that there are <em>"less or equal"</em>

      /// supply/demand constraints in the definition, i.e. the exact

      /// formulation of the problem is the following.

      /**

          \f[ \min\sum_{uv\in A} f(uv) \cdot cost(uv) \f]

          \f[ \sum_{uv\in A} f(uv) - \sum_{vu\in A} f(vu) \leq

              sup(u) \quad \forall u\in V \f]

          \f[ lower(uv) \leq f(uv) \leq upper(uv) \quad \forall uv\in A \f]

      */

      /// It means that the total demand must be less or equal to the 

      /// total supply (i.e. \f$\sum_{u\in V} sup(u)\f$ must be zero or

      /// positive) and all the demands have to be satisfied, but there

      /// could be supplies that are not carried out from the supply

      /// nodes.

      LEQ,

      /// It is just an alias for the \c LEQ option.

      SATISFY_DEMANDS = LEQ

    };

    /// \brief Constants for selecting the pivot rule.

    ///

    /// Enum type containing constants for selecting the pivot rule for

    /// the \ref run() function.

    ///

    /// \ref NetworkSimplex provides five different pivot rule

    /// implementations that significantly affect the running time

    /// of the algorithm.

    /// By default \ref BLOCK_SEARCH "Block Search" is used, which

    /// proved to be the most efficient and the most robust on various

    /// test inputs according to our benchmark tests.

    /// However another pivot rule can be selected using the \ref run()

    /// function with the proper parameter.

    enum PivotRule {

      /// The First Eligible pivot rule.

      /// The next eligible arc is selected in a wraparound fashion

      /// in every iteration.

      FIRST_ELIGIBLE,

      /// The Best Eligible pivot rule.

      /// The best eligible arc is selected in every iteration.

      BEST_ELIGIBLE,

      /// The Block Search pivot rule.

      /// A specified number of arcs are examined in every iteration

      /// in a wraparound fashion and the best eligible arc is selected

      /// from this block.

      BLOCK_SEARCH,

      /// The Candidate List pivot rule.

      /// In a major iteration a candidate list is built from eligible arcs

      /// in a wraparound fashion and in the following minor iterations

      /// the best eligible arc is selected from this list.

      CANDIDATE_LIST,

      /// The Altering Candidate List pivot rule.

      /// It is a modified version of the Candidate List method.

      /// It keeps only the several best eligible arcs from the former

      /// candidate list and extends this list in every iteration.

      ALTERING_LIST

    };

  private:

    TEMPLATE_DIGRAPH_TYPEDEFS(GR);

    typedef typename GR::template ArcMap<Cost> CostArcMap;

    typedef std::vector<Arc> ArcVector;

    typedef std::vector<Node> NodeVector;

    typedef std::vector<int> IntVector;

    typedef std::vector<bool> BoolVector;

    typedef std::vector<Cost> CostVector;

    // State constants for arcs

    enum ArcStateEnum {

      STATE_UPPER = -1,

      STATE_TREE  =  0,

      STATE_LOWER =  1

    };

  private:

    // Data related to the underlying digraph

    const GR &_graph;

    int _node_num;

    int _arc_num;

    // Parameters of the problem

    CostArcMap *_pcost;

    bool _pstsup;

    Node _psource, _ptarget;

    SupplyType _stype;

    // Result maps

    FlowMap *_flow_map;

    PotentialMap *_potential_map;

    bool _local_flow;

    bool _local_potential;

    // Data structures for storing the digraph

    IntNodeMap _node_id;

    ArcVector _arc_ref;

    IntVector _source;

    IntVector _target;

    // Node and arc data

    FlowVector _cap;

    CostVector _cost;

    FlowVector _supply;

    FlowVector _flow;

    CostVector _pi;

    // Data for storing the spanning tree structure

    IntVector _parent;

    IntVector _pred;

    IntVector _thread;

    IntVector _rev_thread;

    IntVector _succ_num;

    IntVector _last_succ;

    IntVector _dirty_revs;

    BoolVector _forward;

    IntVector _state;

    int _root;

    // Temporary data used in the current pivot iteration

    int in_arc, join, u_in, v_in, u_out, v_out;

    int first, second, right, last;

    int stem, par_stem, new_stem;

  public:

    /// \brief Constant for infinite upper bounds (capacities).

    ///

    /// Constant for infinite upper bounds (capacities).

  private:

    // Implementation of the First Eligible pivot rule

    class FirstEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

      // Pivot rule data

      int _next_arc;

    public:

      // Constructor

      FirstEligiblePivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _arc_num(ns._arc_num), _next_arc(0)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        Cost c;

        for (int e = _next_arc; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

            _next_arc = e + 1;

            return true;

          }

        }

        for (int e = 0; e < _next_arc; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < 0) {

            _in_arc = e;

            _next_arc = e + 1;

            return true;

          }

        }

        return false;

      }

    }; //class FirstEligiblePivotRule

    // Implementation of the Best Eligible pivot rule

    class BestEligiblePivotRule

    {

    private:

      // References to the NetworkSimplex class

      const IntVector  &_source;

      const IntVector  &_target;

      const CostVector &_cost;

      const IntVector  &_state;

      const CostVector &_pi;

      int &_in_arc;

      int _arc_num;

    public:

      // Constructor

      BestEligiblePivotRule(NetworkSimplex &ns) :

        _source(ns._source), _target(ns._target),

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _arc_num(ns._arc_num)

      {}

      // Find next entering arc

      bool findEnteringArc() {

        Cost c, min = 0;

        for (int e = 0; e < _arc_num; ++e) {

          c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]);

          if (c < min) {

            min = c;

            _in_arc = e;

          }

        }

        return min < 0;

      }

    }; //class BestEligiblePivotRule

    // Implementation of the Block Search pivot rule

    class BlockSearchPivotRule

    {

        _cost(ns._cost), _state(ns._state), _pi(ns._pi),

        _in_arc(ns.in_arc), _arc_num(ns._arc_num),