/* -*- 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 #include #include #include #include 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. /// \tparam V The value type used for flow amounts, capacity bounds /// and supply values in the algorithm. By default it is \c int. /// \tparam C The value type used for costs and potentials in the /// algorithm. By default it is the same as \c V. /// /// \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. template class NetworkSimplex { public: /// The type of the flow amounts, capacity bounds and supply values typedef V Value; /// The type of the arc costs typedef C Cost; 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 "greater or equal" /// 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 "less or equal" /// 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 std::vector ArcVector; typedef std::vector NodeVector; typedef std::vector IntVector; typedef std::vector BoolVector; typedef std::vector ValueVector; typedef std::vector 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 bool _have_lower; SupplyType _stype; Value _sum_supply; // Data structures for storing the digraph IntNodeMap _node_id; IntArcMap _arc_id; IntVector _source; IntVector _target; // Node and arc data ValueVector _lower; ValueVector _upper; ValueVector _cap; CostVector _cost; ValueVector _supply; ValueVector _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; Value delta; public: /// \brief Constant for infinite upper bounds (capacities). /// /// Constant for infinite upper bounds (capacities). /// It is \c std::numeric_limits::infinity() if available, /// \c std::numeric_limits::max() otherwise. const Value INF; 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 { 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 _block_size; int _next_arc; public: // Constructor BlockSearchPivotRule(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) { // The main parameters of the pivot rule const double BLOCK_SIZE_FACTOR = 2.0; const int MIN_BLOCK_SIZE = 10; _block_size = std::max( int(BLOCK_SIZE_FACTOR * std::sqrt(double(_arc_num))), MIN_BLOCK_SIZE ); } // Find next entering arc bool findEnteringArc() { Cost c, min = 0; int cnt = _block_size; int e, min_arc = _next_arc; for (e = _next_arc; e < _arc_num; ++e) { c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (c < min) { min = c; min_arc = e; } if (--cnt == 0) { if (min < 0) break; cnt = _block_size; } } if (min == 0 || cnt > 0) { for (e = 0; e < _next_arc; ++e) { c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (c < min) { min = c; min_arc = e; } if (--cnt == 0) { if (min < 0) break; cnt = _block_size; } } } if (min >= 0) return false; _in_arc = min_arc; _next_arc = e; return true; } }; //class BlockSearchPivotRule // Implementation of the Candidate List pivot rule class CandidateListPivotRule { 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 IntVector _candidates; int _list_length, _minor_limit; int _curr_length, _minor_count; int _next_arc; public: /// Constructor CandidateListPivotRule(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) { // The main parameters of the pivot rule const double LIST_LENGTH_FACTOR = 1.0; const int MIN_LIST_LENGTH = 10; const double MINOR_LIMIT_FACTOR = 0.1; const int MIN_MINOR_LIMIT = 3; _list_length = std::max( int(LIST_LENGTH_FACTOR * std::sqrt(double(_arc_num))), MIN_LIST_LENGTH ); _minor_limit = std::max( int(MINOR_LIMIT_FACTOR * _list_length), MIN_MINOR_LIMIT ); _curr_length = _minor_count = 0; _candidates.resize(_list_length); } /// Find next entering arc bool findEnteringArc() { Cost min, c; int e, min_arc = _next_arc; if (_curr_length > 0 && _minor_count < _minor_limit) { // Minor iteration: select the best eligible arc from the // current candidate list ++_minor_count; min = 0; for (int i = 0; i < _curr_length; ++i) { e = _candidates[i]; c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (c < min) { min = c; min_arc = e; } if (c >= 0) { _candidates[i--] = _candidates[--_curr_length]; } } if (min < 0) { _in_arc = min_arc; return true; } } // Major iteration: build a new candidate list min = 0; _curr_length = 0; for (e = _next_arc; e < _arc_num; ++e) { c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (c < 0) { _candidates[_curr_length++] = e; if (c < min) { min = c; min_arc = e; } if (_curr_length == _list_length) break; } } if (_curr_length < _list_length) { for (e = 0; e < _next_arc; ++e) { c = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (c < 0) { _candidates[_curr_length++] = e; if (c < min) { min = c; min_arc = e; } if (_curr_length == _list_length) break; } } } if (_curr_length == 0) return false; _minor_count = 1; _in_arc = min_arc; _next_arc = e; return true; } }; //class CandidateListPivotRule // Implementation of the Altering Candidate List pivot rule class AlteringListPivotRule { 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 _block_size, _head_length, _curr_length; int _next_arc; IntVector _candidates; CostVector _cand_cost; // Functor class to compare arcs during sort of the candidate list class SortFunc { private: const CostVector &_map; public: SortFunc(const CostVector &map) : _map(map) {} bool operator()(int left, int right) { return _map[left] > _map[right]; } }; SortFunc _sort_func; public: // Constructor AlteringListPivotRule(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), _cand_cost(ns._arc_num), _sort_func(_cand_cost) { // The main parameters of the pivot rule const double BLOCK_SIZE_FACTOR = 1.5; const int MIN_BLOCK_SIZE = 10; const double HEAD_LENGTH_FACTOR = 0.1; const int MIN_HEAD_LENGTH = 3; _block_size = std::max( int(BLOCK_SIZE_FACTOR * std::sqrt(double(_arc_num))), MIN_BLOCK_SIZE ); _head_length = std::max( int(HEAD_LENGTH_FACTOR * _block_size), MIN_HEAD_LENGTH ); _candidates.resize(_head_length + _block_size); _curr_length = 0; } // Find next entering arc bool findEnteringArc() { // Check the current candidate list int e; for (int i = 0; i < _curr_length; ++i) { e = _candidates[i]; _cand_cost[e] = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (_cand_cost[e] >= 0) { _candidates[i--] = _candidates[--_curr_length]; } } // Extend the list int cnt = _block_size; int last_arc = 0; int limit = _head_length; for (int e = _next_arc; e < _arc_num; ++e) { _cand_cost[e] = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (_cand_cost[e] < 0) { _candidates[_curr_length++] = e; last_arc = e; } if (--cnt == 0) { if (_curr_length > limit) break; limit = 0; cnt = _block_size; } } if (_curr_length <= limit) { for (int e = 0; e < _next_arc; ++e) { _cand_cost[e] = _state[e] * (_cost[e] + _pi[_source[e]] - _pi[_target[e]]); if (_cand_cost[e] < 0) { _candidates[_curr_length++] = e; last_arc = e; } if (--cnt == 0) { if (_curr_length > limit) break; limit = 0; cnt = _block_size; } } } if (_curr_length == 0) return false; _next_arc = last_arc + 1; // Make heap of the candidate list (approximating a partial sort) make_heap( _candidates.begin(), _candidates.begin() + _curr_length, _sort_func ); // Pop the first element of the heap _in_arc = _candidates[0]; pop_heap( _candidates.begin(), _candidates.begin() + _curr_length, _sort_func ); _curr_length = std::min(_head_length, _curr_length - 1); return true; } }; //class AlteringListPivotRule public: /// \brief Constructor. /// /// The constructor of the class. /// /// \param graph The digraph the algorithm runs on. NetworkSimplex(const GR& graph) : _graph(graph), _node_id(graph), _arc_id(graph), INF(std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : std::numeric_limits::max()) { // Check the value types LEMON_ASSERT(std::numeric_limits::is_signed, "The flow type of NetworkSimplex must be signed"); LEMON_ASSERT(std::numeric_limits::is_signed, "The cost type of NetworkSimplex must be signed"); // Resize vectors _node_num = countNodes(_graph); _arc_num = countArcs(_graph); int all_node_num = _node_num + 1; int all_arc_num = _arc_num + _node_num; _source.resize(all_arc_num); _target.resize(all_arc_num); _lower.resize(all_arc_num); _upper.resize(all_arc_num); _cap.resize(all_arc_num); _cost.resize(all_arc_num); _supply.resize(all_node_num); _flow.resize(all_arc_num); _pi.resize(all_node_num); _parent.resize(all_node_num); _pred.resize(all_node_num); _forward.resize(all_node_num); _thread.resize(all_node_num); _rev_thread.resize(all_node_num); _succ_num.resize(all_node_num); _last_succ.resize(all_node_num); _state.resize(all_arc_num); // Copy the graph (store the arcs in a mixed order) int i = 0; for (NodeIt n(_graph); n != INVALID; ++n, ++i) { _node_id[n] = i; } int k = std::max(int(std::sqrt(double(_arc_num))), 10); i = 0; for (ArcIt a(_graph); a != INVALID; ++a) { _arc_id[a] = i; _source[i] = _node_id[_graph.source(a)]; _target[i] = _node_id[_graph.target(a)]; if ((i += k) >= _arc_num) i = (i % k) + 1; } // Initialize maps for (int i = 0; i != _node_num; ++i) { _supply[i] = 0; } for (int i = 0; i != _arc_num; ++i) { _lower[i] = 0; _upper[i] = INF; _cost[i] = 1; } _have_lower = false; _stype = GEQ; } /// \name Parameters /// The parameters of the algorithm can be specified using these /// functions. /// @{ /// \brief Set the lower bounds on the arcs. /// /// This function sets the lower bounds on the arcs. /// If it is not used before calling \ref run(), the lower bounds /// will be set to zero on all arcs. /// /// \param map An arc map storing the lower bounds. /// Its \c Value type must be convertible to the \c Value type /// of the algorithm. /// /// \return (*this) template NetworkSimplex& lowerMap(const LowerMap& map) { _have_lower = true; for (ArcIt a(_graph); a != INVALID; ++a) { _lower[_arc_id[a]] = map[a]; } return *this; } /// \brief Set the upper bounds (capacities) on the arcs. /// /// This function sets the upper bounds (capacities) on the arcs. /// If it is not used before calling \ref run(), the upper bounds /// will be set to \ref INF on all arcs (i.e. the flow value will be /// unbounded from above on each arc). /// /// \param map An arc map storing the upper bounds. /// Its \c Value type must be convertible to the \c Value type /// of the algorithm. /// /// \return (*this) template NetworkSimplex& upperMap(const UpperMap& map) { for (ArcIt a(_graph); a != INVALID; ++a) { _upper[_arc_id[a]] = map[a]; } return *this; } /// \brief Set the costs of the arcs. /// /// This function sets the costs of the arcs. /// If it is not used before calling \ref run(), the costs /// will be set to \c 1 on all arcs. /// /// \param map An arc map storing the costs. /// Its \c Value type must be convertible to the \c Cost type /// of the algorithm. /// /// \return (*this) template NetworkSimplex& costMap(const CostMap& map) { for (ArcIt a(_graph); a != INVALID; ++a) { _cost[_arc_id[a]] = map[a]; } return *this; } /// \brief Set the supply values of the nodes. /// /// This function sets the supply values of the nodes. /// If neither this function nor \ref stSupply() is used before /// calling \ref run(), the supply of each node will be set to zero. /// (It makes sense only if non-zero lower bounds are given.) /// /// \param map A node map storing the supply values. /// Its \c Value type must be convertible to the \c Value type /// of the algorithm. /// /// \return (*this) template NetworkSimplex& supplyMap(const SupplyMap& map) { for (NodeIt n(_graph); n != INVALID; ++n) { _supply[_node_id[n]] = map[n]; } return *this; } /// \brief Set single source and target nodes and a supply value. /// /// This function sets a single source node and a single target node /// and the required flow value. /// If neither this function nor \ref supplyMap() is used before /// calling \ref run(), the supply of each node will be set to zero. /// (It makes sense only if non-zero lower bounds are given.) /// /// Using this function has the same effect as using \ref supplyMap() /// with such a map in which \c k is assigned to \c s, \c -k is /// assigned to \c t and all other nodes have zero supply value. /// /// \param s The source node. /// \param t The target node. /// \param k 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). /// /// \return (*this) NetworkSimplex& stSupply(const Node& s, const Node& t, Value k) { for (int i = 0; i != _node_num; ++i) { _supply[i] = 0; } _supply[_node_id[s]] = k; _supply[_node_id[t]] = -k; return *this; } /// \brief Set the type of the supply constraints. /// /// This function sets the type of the supply/demand constraints. /// If it is not used before calling \ref run(), the \ref GEQ supply /// type will be used. /// /// For more information see \ref SupplyType. /// /// \return (*this) NetworkSimplex& supplyType(SupplyType supply_type) { _stype = supply_type; return *this; } /// @} /// \name Execution Control /// The algorithm can be executed using \ref run(). /// @{ /// \brief Run the algorithm. /// /// This function runs the algorithm. /// The paramters can be specified using functions \ref lowerMap(), /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(), /// \ref supplyType(). /// For example, /// \code /// NetworkSimplex ns(graph); /// ns.lowerMap(lower).upperMap(upper).costMap(cost) /// .supplyMap(sup).run(); /// \endcode /// /// This function can be called more than once. All the parameters /// that have been given are kept for the next call, unless /// \ref reset() is called, thus only the modified parameters /// have to be set again. See \ref reset() for examples. /// However the underlying digraph must not be modified after this /// class have been constructed, since it copies and extends the graph. /// /// \param pivot_rule The pivot rule that will be used during the /// algorithm. For more information see \ref PivotRule. /// /// \return \c INFEASIBLE if no feasible flow exists, /// \n \c OPTIMAL if 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), /// \n \c UNBOUNDED if the objective function of the problem is /// unbounded, i.e. there is a directed cycle having negative total /// cost and infinite upper bound. /// /// \see ProblemType, PivotRule ProblemType run(PivotRule pivot_rule = BLOCK_SEARCH) { if (!init()) return INFEASIBLE; return start(pivot_rule); } /// \brief Reset all the parameters that have been given before. /// /// This function resets all the paramaters that have been given /// before using functions \ref lowerMap(), \ref upperMap(), /// \ref costMap(), \ref supplyMap(), \ref stSupply(), \ref supplyType(). /// /// It is useful for multiple run() calls. If this function is not /// used, all the parameters given before are kept for the next /// \ref run() call. /// However the underlying digraph must not be modified after this /// class have been constructed, since it copies and extends the graph. /// /// For example, /// \code /// NetworkSimplex ns(graph); /// /// // First run /// ns.lowerMap(lower).upperMap(upper).costMap(cost) /// .supplyMap(sup).run(); /// /// // Run again with modified cost map (reset() is not called, /// // so only the cost map have to be set again) /// cost[e] += 100; /// ns.costMap(cost).run(); /// /// // Run again from scratch using reset() /// // (the lower bounds will be set to zero on all arcs) /// ns.reset(); /// ns.upperMap(capacity).costMap(cost) /// .supplyMap(sup).run(); /// \endcode /// /// \return (*this) NetworkSimplex& reset() { for (int i = 0; i != _node_num; ++i) { _supply[i] = 0; } for (int i = 0; i != _arc_num; ++i) { _lower[i] = 0; _upper[i] = INF; _cost[i] = 1; } _have_lower = false; _stype = GEQ; return *this; } /// @} /// \name Query Functions /// The results of the algorithm can be obtained using these /// functions.\n /// The \ref run() function must be called before using them. /// @{ /// \brief Return the total cost of the found flow. /// /// This function returns the total cost of the found flow. /// Its complexity is O(e). /// /// \note The return type of the function can be specified as a /// template parameter. For example, /// \code /// ns.totalCost(); /// \endcode /// It is useful if the total cost cannot be stored in the \c Cost /// type of the algorithm, which is the default return type of the /// function. /// /// \pre \ref run() must be called before using this function. template Number totalCost() const { Number c = 0; for (ArcIt a(_graph); a != INVALID; ++a) { int i = _arc_id[a]; c += Number(_flow[i]) * Number(_cost[i]); } return c; } #ifndef DOXYGEN Cost totalCost() const { return totalCost(); } #endif /// \brief Return the flow on the given arc. /// /// This function returns the flow on the given arc. /// /// \pre \ref run() must be called before using this function. Value flow(const Arc& a) const { return _flow[_arc_id[a]]; } /// \brief Return the flow map (the primal solution). /// /// This function copies the flow value on each arc into the given /// map. The \c Value type of the algorithm must be convertible to /// the \c Value type of the map. /// /// \pre \ref run() must be called before using this function. template void flowMap(FlowMap &map) const { for (ArcIt a(_graph); a != INVALID; ++a) { map.set(a, _flow[_arc_id[a]]); } } /// \brief Return the potential (dual value) of the given node. /// /// This function returns the potential (dual value) of the /// given node. /// /// \pre \ref run() must be called before using this function. Cost potential(const Node& n) const { return _pi[_node_id[n]]; } /// \brief Return the potential map (the dual solution). /// /// This function copies the potential (dual value) of each node /// into the given map. /// The \c Cost type of the algorithm must be convertible to the /// \c Value type of the map. /// /// \pre \ref run() must be called before using this function. template void potentialMap(PotentialMap &map) const { for (NodeIt n(_graph); n != INVALID; ++n) { map.set(n, _pi[_node_id[n]]); } } /// @} private: // Initialize internal data structures bool init() { if (_node_num == 0) return false; // Check the sum of supply values _sum_supply = 0; for (int i = 0; i != _node_num; ++i) { _sum_supply += _supply[i]; } if ( !((_stype == GEQ && _sum_supply <= 0) || (_stype == LEQ && _sum_supply >= 0)) ) return false; // Remove non-zero lower bounds if (_have_lower) { for (int i = 0; i != _arc_num; ++i) { Value c = _lower[i]; if (c >= 0) { _cap[i] = _upper[i] < INF ? _upper[i] - c : INF; } else { _cap[i] = _upper[i] < INF + c ? _upper[i] - c : INF; } _supply[_source[i]] -= c; _supply[_target[i]] += c; } } else { for (int i = 0; i != _arc_num; ++i) { _cap[i] = _upper[i]; } } // Initialize artifical cost Cost ART_COST; if (std::numeric_limits::is_exact) { ART_COST = std::numeric_limits::max() / 4 + 1; } else { ART_COST = std::numeric_limits::min(); for (int i = 0; i != _arc_num; ++i) { if (_cost[i] > ART_COST) ART_COST = _cost[i]; } ART_COST = (ART_COST + 1) * _node_num; } // Initialize arc maps for (int i = 0; i != _arc_num; ++i) { _flow[i] = 0; _state[i] = STATE_LOWER; } // Set data for the artificial root node _root = _node_num; _parent[_root] = -1; _pred[_root] = -1; _thread[_root] = 0; _rev_thread[0] = _root; _succ_num[_root] = _node_num + 1; _last_succ[_root] = _root - 1; _supply[_root] = -_sum_supply; _pi[_root] = _sum_supply < 0 ? -ART_COST : ART_COST; // Add artificial arcs and initialize the spanning tree data structure for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { _parent[u] = _root; _pred[u] = e; _thread[u] = u + 1; _rev_thread[u + 1] = u; _succ_num[u] = 1; _last_succ[u] = u; _cost[e] = ART_COST; _cap[e] = INF; _state[e] = STATE_TREE; if (_supply[u] > 0 || (_supply[u] == 0 && _sum_supply <= 0)) { _flow[e] = _supply[u]; _forward[u] = true; _pi[u] = -ART_COST + _pi[_root]; } else { _flow[e] = -_supply[u]; _forward[u] = false; _pi[u] = ART_COST + _pi[_root]; } } return true; } // Find the join node void findJoinNode() { int u = _source[in_arc]; int v = _target[in_arc]; while (u != v) { if (_succ_num[u] < _succ_num[v]) { u = _parent[u]; } else { v = _parent[v]; } } join = u; } // Find the leaving arc of the cycle and returns true if the // leaving arc is not the same as the entering arc bool findLeavingArc() { // Initialize first and second nodes according to the direction // of the cycle if (_state[in_arc] == STATE_LOWER) { first = _source[in_arc]; second = _target[in_arc]; } else { first = _target[in_arc]; second = _source[in_arc]; } delta = _cap[in_arc]; int result = 0; Value d; int e; // Search the cycle along the path form the first node to the root for (int u = first; u != join; u = _parent[u]) { e = _pred[u]; d = _forward[u] ? _flow[e] : (_cap[e] == INF ? INF : _cap[e] - _flow[e]); if (d < delta) { delta = d; u_out = u; result = 1; } } // Search the cycle along the path form the second node to the root for (int u = second; u != join; u = _parent[u]) { e = _pred[u]; d = _forward[u] ? (_cap[e] == INF ? INF : _cap[e] - _flow[e]) : _flow[e]; if (d <= delta) { delta = d; u_out = u; result = 2; } } if (result == 1) { u_in = first; v_in = second; } else { u_in = second; v_in = first; } return result != 0; } // Change _flow and _state vectors void changeFlow(bool change) { // Augment along the cycle if (delta > 0) { Value val = _state[in_arc] * delta; _flow[in_arc] += val; for (int u = _source[in_arc]; u != join; u = _parent[u]) { _flow[_pred[u]] += _forward[u] ? -val : val; } for (int u = _target[in_arc]; u != join; u = _parent[u]) { _flow[_pred[u]] += _forward[u] ? val : -val; } } // Update the state of the entering and leaving arcs if (change) { _state[in_arc] = STATE_TREE; _state[_pred[u_out]] = (_flow[_pred[u_out]] == 0) ? STATE_LOWER : STATE_UPPER; } else { _state[in_arc] = -_state[in_arc]; } } // Update the tree structure void updateTreeStructure() { int u, w; int old_rev_thread = _rev_thread[u_out]; int old_succ_num = _succ_num[u_out]; int old_last_succ = _last_succ[u_out]; v_out = _parent[u_out]; u = _last_succ[u_in]; // the last successor of u_in right = _thread[u]; // the node after it // Handle the case when old_rev_thread equals to v_in // (it also means that join and v_out coincide) if (old_rev_thread == v_in) { last = _thread[_last_succ[u_out]]; } else { last = _thread[v_in]; } // Update _thread and _parent along the stem nodes (i.e. the nodes // between u_in and u_out, whose parent have to be changed) _thread[v_in] = stem = u_in; _dirty_revs.clear(); _dirty_revs.push_back(v_in); par_stem = v_in; while (stem != u_out) { // Insert the next stem node into the thread list new_stem = _parent[stem]; _thread[u] = new_stem; _dirty_revs.push_back(u); // Remove the subtree of stem from the thread list w = _rev_thread[stem]; _thread[w] = right; _rev_thread[right] = w; // Change the parent node and shift stem nodes _parent[stem] = par_stem; par_stem = stem; stem = new_stem; // Update u and right u = _last_succ[stem] == _last_succ[par_stem] ? _rev_thread[par_stem] : _last_succ[stem]; right = _thread[u]; } _parent[u_out] = par_stem; _thread[u] = last; _rev_thread[last] = u; _last_succ[u_out] = u; // Remove the subtree of u_out from the thread list except for // the case when old_rev_thread equals to v_in // (it also means that join and v_out coincide) if (old_rev_thread != v_in) { _thread[old_rev_thread] = right; _rev_thread[right] = old_rev_thread; } // Update _rev_thread using the new _thread values for (int i = 0; i < int(_dirty_revs.size()); ++i) { u = _dirty_revs[i]; _rev_thread[_thread[u]] = u; } // Update _pred, _forward, _last_succ and _succ_num for the // stem nodes from u_out to u_in int tmp_sc = 0, tmp_ls = _last_succ[u_out]; u = u_out; while (u != u_in) { w = _parent[u]; _pred[u] = _pred[w]; _forward[u] = !_forward[w]; tmp_sc += _succ_num[u] - _succ_num[w]; _succ_num[u] = tmp_sc; _last_succ[w] = tmp_ls; u = w; } _pred[u_in] = in_arc; _forward[u_in] = (u_in == _source[in_arc]); _succ_num[u_in] = old_succ_num; // Set limits for updating _last_succ form v_in and v_out // towards the root int up_limit_in = -1; int up_limit_out = -1; if (_last_succ[join] == v_in) { up_limit_out = join; } else { up_limit_in = join; } // Update _last_succ from v_in towards the root for (u = v_in; u != up_limit_in && _last_succ[u] == v_in; u = _parent[u]) { _last_succ[u] = _last_succ[u_out]; } // Update _last_succ from v_out towards the root if (join != old_rev_thread && v_in != old_rev_thread) { for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; u = _parent[u]) { _last_succ[u] = old_rev_thread; } } else { for (u = v_out; u != up_limit_out && _last_succ[u] == old_last_succ; u = _parent[u]) { _last_succ[u] = _last_succ[u_out]; } } // Update _succ_num from v_in to join for (u = v_in; u != join; u = _parent[u]) { _succ_num[u] += old_succ_num; } // Update _succ_num from v_out to join for (u = v_out; u != join; u = _parent[u]) { _succ_num[u] -= old_succ_num; } } // Update potentials void updatePotential() { Cost sigma = _forward[u_in] ? _pi[v_in] - _pi[u_in] - _cost[_pred[u_in]] : _pi[v_in] - _pi[u_in] + _cost[_pred[u_in]]; // Update potentials in the subtree, which has been moved int end = _thread[_last_succ[u_in]]; for (int u = u_in; u != end; u = _thread[u]) { _pi[u] += sigma; } } // Execute the algorithm ProblemType start(PivotRule pivot_rule) { // Select the pivot rule implementation switch (pivot_rule) { case FIRST_ELIGIBLE: return start(); case BEST_ELIGIBLE: return start(); case BLOCK_SEARCH: return start(); case CANDIDATE_LIST: return start(); case ALTERING_LIST: return start(); } return INFEASIBLE; // avoid warning } template ProblemType start() { PivotRuleImpl pivot(*this); // Execute the Network Simplex algorithm while (pivot.findEnteringArc()) { findJoinNode(); bool change = findLeavingArc(); if (delta >= INF) return UNBOUNDED; changeFlow(change); if (change) { updateTreeStructure(); updatePotential(); } } // Check feasibility if (_sum_supply < 0) { for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { if (_supply[u] >= 0 && _flow[e] != 0) return INFEASIBLE; } } else if (_sum_supply > 0) { for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { if (_supply[u] <= 0 && _flow[e] != 0) return INFEASIBLE; } } else { for (int u = 0, e = _arc_num; u != _node_num; ++u, ++e) { if (_flow[e] != 0) return INFEASIBLE; } } // Transform the solution and the supply map to the original form if (_have_lower) { for (int i = 0; i != _arc_num; ++i) { Value c = _lower[i]; if (c != 0) { _flow[i] += c; _supply[_source[i]] += c; _supply[_target[i]] -= c; } } } return OPTIMAL; } }; //class NetworkSimplex ///@} } //namespace lemon #endif //LEMON_NETWORK_SIMPLEX_H