/* -*- mode: C++; indent-tabs-mode: nil; -*- * * This file is a part of LEMON, a generic C++ optimization library. * * Copyright (C) 2003-2013 * 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_algs /// \file /// \brief Cycle-canceling algorithms for finding a minimum cost flow. #include #include #include #include #include #include #include #include #include #include #include #include namespace lemon { /// \addtogroup min_cost_flow_algs /// @{ /// \brief Implementation of cycle-canceling algorithms for /// finding a \ref min_cost_flow "minimum cost flow". /// /// \ref CycleCanceling implements three different cycle-canceling /// algorithms for finding a \ref min_cost_flow "minimum cost flow" /// \cite amo93networkflows, \cite klein67primal, /// \cite goldberg89cyclecanceling. /// The most efficent one is the \ref CANCEL_AND_TIGHTEN /// "Cancel-and-Tighten" algorithm, thus it is the default method. /// It runs in strongly polynomial time \f$O(n^2 m^2 \log n)\f$, /// but in practice, it is typically orders of magnitude slower than /// the scaling algorithms and \ref NetworkSimplex. /// (For more information, see \ref min_cost_flow_algs "the module page".) /// /// 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 number type used for flow amounts, capacity bounds /// and supply values in the algorithm. By default, it is \c int. /// \tparam C The number type used for costs and potentials in the /// algorithm. By default, it is the same as \c V. /// /// \warning Both \c V and \c C must be signed number types. /// \warning All input data (capacities, supply values, and costs) must /// be integer. /// \warning This algorithm does not support negative costs for /// arcs having infinite upper bound. /// /// \note For more information about the three available methods, /// see \ref Method. #ifdef DOXYGEN template #else template #endif class CycleCanceling { public: /// The type of the digraph typedef GR Digraph; /// 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 digraph contains an arc of negative cost and infinite /// upper bound. It means that the objective function is unbounded /// on that arc, however, note that it could actually be bounded /// over the feasible flows, but this algroithm cannot handle /// these cases. UNBOUNDED }; /// \brief Constants for selecting the used method. /// /// Enum type containing constants for selecting the used method /// for the \ref run() function. /// /// \ref CycleCanceling provides three different cycle-canceling /// methods. By default, \ref CANCEL_AND_TIGHTEN "Cancel-and-Tighten" /// is used, which is by far the most efficient and the most robust. /// However, the other methods can be selected using the \ref run() /// function with the proper parameter. enum Method { /// A simple cycle-canceling method, which uses the /// \ref BellmanFord "Bellman-Ford" algorithm for detecting negative /// cycles in the residual network. /// The number of Bellman-Ford iterations is bounded by a successively /// increased limit. SIMPLE_CYCLE_CANCELING, /// The "Minimum Mean Cycle-Canceling" algorithm, which is a /// well-known strongly polynomial method /// \cite goldberg89cyclecanceling. It improves along a /// \ref min_mean_cycle "minimum mean cycle" in each iteration. /// Its running time complexity is \f$O(n^2 m^3 \log n)\f$. MINIMUM_MEAN_CYCLE_CANCELING, /// The "Cancel-and-Tighten" algorithm, which can be viewed as an /// improved version of the previous method /// \cite goldberg89cyclecanceling. /// It is faster both in theory and in practice, its running time /// complexity is \f$O(n^2 m^2 \log n)\f$. CANCEL_AND_TIGHTEN }; private: TEMPLATE_DIGRAPH_TYPEDEFS(GR); typedef std::vector IntVector; typedef std::vector DoubleVector; typedef std::vector ValueVector; typedef std::vector CostVector; typedef std::vector BoolVector; // Note: vector is used instead of vector for efficiency reasons private: template class StaticVectorMap { public: typedef KT Key; typedef VT Value; StaticVectorMap(std::vector& v) : _v(v) {} const Value& operator[](const Key& key) const { return _v[StaticDigraph::id(key)]; } Value& operator[](const Key& key) { return _v[StaticDigraph::id(key)]; } void set(const Key& key, const Value& val) { _v[StaticDigraph::id(key)] = val; } private: std::vector& _v; }; typedef StaticVectorMap CostNodeMap; typedef StaticVectorMap CostArcMap; private: // Data related to the underlying digraph const GR &_graph; int _node_num; int _arc_num; int _res_node_num; int _res_arc_num; int _root; // Parameters of the problem bool _has_lower; Value _sum_supply; // Data structures for storing the digraph IntNodeMap _node_id; IntArcMap _arc_idf; IntArcMap _arc_idb; IntVector _first_out; BoolVector _forward; IntVector _source; IntVector _target; IntVector _reverse; // Node and arc data ValueVector _lower; ValueVector _upper; CostVector _cost; ValueVector _supply; ValueVector _res_cap; CostVector _pi; // Data for a StaticDigraph structure typedef std::pair IntPair; StaticDigraph _sgr; std::vector _arc_vec; std::vector _cost_vec; IntVector _id_vec; CostArcMap _cost_map; CostNodeMap _pi_map; 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; public: /// \brief Constructor. /// /// The constructor of the class. /// /// \param graph The digraph the algorithm runs on. CycleCanceling(const GR& graph) : _graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph), _cost_map(_cost_vec), _pi_map(_pi), INF(std::numeric_limits::has_infinity ? std::numeric_limits::infinity() : std::numeric_limits::max()) { // Check the number types LEMON_ASSERT(std::numeric_limits::is_signed, "The flow type of CycleCanceling must be signed"); LEMON_ASSERT(std::numeric_limits::is_signed, "The cost type of CycleCanceling must be signed"); // Reset data structures reset(); } /// \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 CycleCanceling& lowerMap(const LowerMap& map) { _has_lower = true; for (ArcIt a(_graph); a != INVALID; ++a) { _lower[_arc_idf[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). /// /// \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 CycleCanceling& upperMap(const UpperMap& map) { for (ArcIt a(_graph); a != INVALID; ++a) { _upper[_arc_idf[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 CycleCanceling& costMap(const CostMap& map) { for (ArcIt a(_graph); a != INVALID; ++a) { _cost[_arc_idf[a]] = map[a]; _cost[_arc_idb[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. /// /// \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 CycleCanceling& 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. /// /// Using this function has the same effect as using \ref supplyMap() /// with 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) CycleCanceling& stSupply(const Node& s, const Node& t, Value k) { for (int i = 0; i != _res_node_num; ++i) { _supply[i] = 0; } _supply[_node_id[s]] = k; _supply[_node_id[t]] = -k; 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(). /// For example, /// \code /// CycleCanceling cc(graph); /// cc.lowerMap(lower).upperMap(upper).costMap(cost) /// .supplyMap(sup).run(); /// \endcode /// /// This function can be called more than once. All the given parameters /// are kept for the next call, unless \ref resetParams() or \ref reset() /// is used, thus only the modified parameters have to be set again. /// If the underlying digraph was also modified after the construction /// of the class (or the last \ref reset() call), then the \ref reset() /// function must be called. /// /// \param method The cycle-canceling method that will be used. /// For more information, see \ref Method. /// /// \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 digraph contains an arc of negative cost /// and infinite upper bound. It means that the objective function /// is unbounded on that arc, however, note that it could actually be /// bounded over the feasible flows, but this algroithm cannot handle /// these cases. /// /// \see ProblemType, Method /// \see resetParams(), reset() ProblemType run(Method method = CANCEL_AND_TIGHTEN) { ProblemType pt = init(); if (pt != OPTIMAL) return pt; start(method); return OPTIMAL; } /// \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(). /// /// It is useful for multiple \ref run() calls. Basically, all the given /// parameters are kept for the next \ref run() call, unless /// \ref resetParams() or \ref reset() is used. /// If the underlying digraph was also modified after the construction /// of the class or the last \ref reset() call, then the \ref reset() /// function must be used, otherwise \ref resetParams() is sufficient. /// /// For example, /// \code /// CycleCanceling cs(graph); /// /// // First run /// cc.lowerMap(lower).upperMap(upper).costMap(cost) /// .supplyMap(sup).run(); /// /// // Run again with modified cost map (resetParams() is not called, /// // so only the cost map have to be set again) /// cost[e] += 100; /// cc.costMap(cost).run(); /// /// // Run again from scratch using resetParams() /// // (the lower bounds will be set to zero on all arcs) /// cc.resetParams(); /// cc.upperMap(capacity).costMap(cost) /// .supplyMap(sup).run(); /// \endcode /// /// \return (*this) /// /// \see reset(), run() CycleCanceling& resetParams() { for (int i = 0; i != _res_node_num; ++i) { _supply[i] = 0; } int limit = _first_out[_root]; for (int j = 0; j != limit; ++j) { _lower[j] = 0; _upper[j] = INF; _cost[j] = _forward[j] ? 1 : -1; } for (int j = limit; j != _res_arc_num; ++j) { _lower[j] = 0; _upper[j] = INF; _cost[j] = 0; _cost[_reverse[j]] = 0; } _has_lower = false; return *this; } /// \brief Reset the internal data structures and all the parameters /// that have been given before. /// /// This function resets the internal data structures and all the /// paramaters that have been given before using functions \ref lowerMap(), /// \ref upperMap(), \ref costMap(), \ref supplyMap(), \ref stSupply(). /// /// It is useful for multiple \ref run() calls. Basically, all the given /// parameters are kept for the next \ref run() call, unless /// \ref resetParams() or \ref reset() is used. /// If the underlying digraph was also modified after the construction /// of the class or the last \ref reset() call, then the \ref reset() /// function must be used, otherwise \ref resetParams() is sufficient. /// /// See \ref resetParams() for examples. /// /// \return (*this) /// /// \see resetParams(), run() CycleCanceling& reset() { // Resize vectors _node_num = countNodes(_graph); _arc_num = countArcs(_graph); _res_node_num = _node_num + 1; _res_arc_num = 2 * (_arc_num + _node_num); _root = _node_num; _first_out.resize(_res_node_num + 1); _forward.resize(_res_arc_num); _source.resize(_res_arc_num); _target.resize(_res_arc_num); _reverse.resize(_res_arc_num); _lower.resize(_res_arc_num); _upper.resize(_res_arc_num); _cost.resize(_res_arc_num); _supply.resize(_res_node_num); _res_cap.resize(_res_arc_num); _pi.resize(_res_node_num); _arc_vec.reserve(_res_arc_num); _cost_vec.reserve(_res_arc_num); _id_vec.reserve(_res_arc_num); // Copy the graph int i = 0, j = 0, k = 2 * _arc_num + _node_num; for (NodeIt n(_graph); n != INVALID; ++n, ++i) { _node_id[n] = i; } i = 0; for (NodeIt n(_graph); n != INVALID; ++n, ++i) { _first_out[i] = j; for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) { _arc_idf[a] = j; _forward[j] = true; _source[j] = i; _target[j] = _node_id[_graph.runningNode(a)]; } for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) { _arc_idb[a] = j; _forward[j] = false; _source[j] = i; _target[j] = _node_id[_graph.runningNode(a)]; } _forward[j] = false; _source[j] = i; _target[j] = _root; _reverse[j] = k; _forward[k] = true; _source[k] = _root; _target[k] = i; _reverse[k] = j; ++j; ++k; } _first_out[i] = j; _first_out[_res_node_num] = k; for (ArcIt a(_graph); a != INVALID; ++a) { int fi = _arc_idf[a]; int bi = _arc_idb[a]; _reverse[fi] = bi; _reverse[bi] = fi; } // Reset parameters resetParams(); 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(m). /// /// \note The return type of the function can be specified as a /// template parameter. For example, /// \code /// cc.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_idb[a]; c += static_cast(_res_cap[i]) * (-static_cast(_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 _res_cap[_arc_idb[a]]; } /// \brief Copy the flow values (the primal solution) into the /// given map. /// /// 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, _res_cap[_arc_idb[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 static_cast(_pi[_node_id[n]]); } /// \brief Copy the potential values (the dual solution) into the /// given map. /// /// 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, static_cast(_pi[_node_id[n]])); } } /// @} private: // Initialize the algorithm ProblemType init() { if (_res_node_num <= 1) return INFEASIBLE; // Check the sum of supply values _sum_supply = 0; for (int i = 0; i != _root; ++i) { _sum_supply += _supply[i]; } if (_sum_supply > 0) return INFEASIBLE; // Check lower and upper bounds LEMON_DEBUG(checkBoundMaps(), "Upper bounds must be greater or equal to the lower bounds"); // Initialize vectors for (int i = 0; i != _res_node_num; ++i) { _pi[i] = 0; } ValueVector excess(_supply); // Remove infinite upper bounds and check negative arcs const Value MAX = std::numeric_limits::max(); int last_out; if (_has_lower) { for (int i = 0; i != _root; ++i) { last_out = _first_out[i+1]; for (int j = _first_out[i]; j != last_out; ++j) { if (_forward[j]) { Value c = _cost[j] < 0 ? _upper[j] : _lower[j]; if (c >= MAX) return UNBOUNDED; excess[i] -= c; excess[_target[j]] += c; } } } } else { for (int i = 0; i != _root; ++i) { last_out = _first_out[i+1]; for (int j = _first_out[i]; j != last_out; ++j) { if (_forward[j] && _cost[j] < 0) { Value c = _upper[j]; if (c >= MAX) return UNBOUNDED; excess[i] -= c; excess[_target[j]] += c; } } } } Value ex, max_cap = 0; for (int i = 0; i != _res_node_num; ++i) { ex = excess[i]; if (ex < 0) max_cap -= ex; } for (int j = 0; j != _res_arc_num; ++j) { if (_upper[j] >= MAX) _upper[j] = max_cap; } // Initialize maps for Circulation and remove non-zero lower bounds ConstMap low(0); typedef typename Digraph::template ArcMap ValueArcMap; typedef typename Digraph::template NodeMap ValueNodeMap; ValueArcMap cap(_graph), flow(_graph); ValueNodeMap sup(_graph); for (NodeIt n(_graph); n != INVALID; ++n) { sup[n] = _supply[_node_id[n]]; } if (_has_lower) { for (ArcIt a(_graph); a != INVALID; ++a) { int j = _arc_idf[a]; Value c = _lower[j]; cap[a] = _upper[j] - c; sup[_graph.source(a)] -= c; sup[_graph.target(a)] += c; } } else { for (ArcIt a(_graph); a != INVALID; ++a) { cap[a] = _upper[_arc_idf[a]]; } } // Find a feasible flow using Circulation Circulation, ValueArcMap, ValueNodeMap> circ(_graph, low, cap, sup); if (!circ.flowMap(flow).run()) return INFEASIBLE; // Set residual capacities and handle GEQ supply type if (_sum_supply < 0) { for (ArcIt a(_graph); a != INVALID; ++a) { Value fa = flow[a]; _res_cap[_arc_idf[a]] = cap[a] - fa; _res_cap[_arc_idb[a]] = fa; sup[_graph.source(a)] -= fa; sup[_graph.target(a)] += fa; } for (NodeIt n(_graph); n != INVALID; ++n) { excess[_node_id[n]] = sup[n]; } for (int a = _first_out[_root]; a != _res_arc_num; ++a) { int u = _target[a]; int ra = _reverse[a]; _res_cap[a] = -_sum_supply + 1; _res_cap[ra] = -excess[u]; _cost[a] = 0; _cost[ra] = 0; } } else { for (ArcIt a(_graph); a != INVALID; ++a) { Value fa = flow[a]; _res_cap[_arc_idf[a]] = cap[a] - fa; _res_cap[_arc_idb[a]] = fa; } for (int a = _first_out[_root]; a != _res_arc_num; ++a) { int ra = _reverse[a]; _res_cap[a] = 1; _res_cap[ra] = 0; _cost[a] = 0; _cost[ra] = 0; } } return OPTIMAL; } // Check if the upper bound is greater than or equal to the lower bound // on each forward arc. bool checkBoundMaps() { for (int j = 0; j != _res_arc_num; ++j) { if (_forward[j] && _upper[j] < _lower[j]) return false; } return true; } // Build a StaticDigraph structure containing the current // residual network void buildResidualNetwork() { _arc_vec.clear(); _cost_vec.clear(); _id_vec.clear(); for (int j = 0; j != _res_arc_num; ++j) { if (_res_cap[j] > 0) { _arc_vec.push_back(IntPair(_source[j], _target[j])); _cost_vec.push_back(_cost[j]); _id_vec.push_back(j); } } _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end()); } // Execute the algorithm and transform the results void start(Method method) { // Execute the algorithm switch (method) { case SIMPLE_CYCLE_CANCELING: startSimpleCycleCanceling(); break; case MINIMUM_MEAN_CYCLE_CANCELING: startMinMeanCycleCanceling(); break; case CANCEL_AND_TIGHTEN: startCancelAndTighten(); break; } // Compute node potentials if (method != SIMPLE_CYCLE_CANCELING) { buildResidualNetwork(); typename BellmanFord ::template SetDistMap::Create bf(_sgr, _cost_map); bf.distMap(_pi_map); bf.init(0); bf.start(); } // Handle non-zero lower bounds if (_has_lower) { int limit = _first_out[_root]; for (int j = 0; j != limit; ++j) { if (_forward[j]) _res_cap[_reverse[j]] += _lower[j]; } } } // Execute the "Simple Cycle Canceling" method void startSimpleCycleCanceling() { // Constants for computing the iteration limits const int BF_FIRST_LIMIT = 2; const double BF_LIMIT_FACTOR = 1.5; typedef StaticVectorMap FilterMap; typedef FilterArcs ResDigraph; typedef StaticVectorMap PredMap; typedef typename BellmanFord ::template SetDistMap ::template SetPredMap::Create BF; // Build the residual network _arc_vec.clear(); _cost_vec.clear(); for (int j = 0; j != _res_arc_num; ++j) { _arc_vec.push_back(IntPair(_source[j], _target[j])); _cost_vec.push_back(_cost[j]); } _sgr.build(_res_node_num, _arc_vec.begin(), _arc_vec.end()); FilterMap filter_map(_res_cap); ResDigraph rgr(_sgr, filter_map); std::vector cycle; std::vector pred(_res_arc_num); PredMap pred_map(pred); BF bf(rgr, _cost_map); bf.distMap(_pi_map).predMap(pred_map); int length_bound = BF_FIRST_LIMIT; bool optimal = false; while (!optimal) { bf.init(0); int iter_num = 0; bool cycle_found = false; while (!cycle_found) { // Perform some iterations of the Bellman-Ford algorithm int curr_iter_num = iter_num + length_bound <= _node_num ? length_bound : _node_num - iter_num; iter_num += curr_iter_num; int real_iter_num = curr_iter_num; for (int i = 0; i < curr_iter_num; ++i) { if (bf.processNextWeakRound()) { real_iter_num = i; break; } } if (real_iter_num < curr_iter_num) { // Optimal flow is found optimal = true; break; } else { // Search for node disjoint negative cycles std::vector state(_res_node_num, 0); int id = 0; for (int u = 0; u != _res_node_num; ++u) { if (state[u] != 0) continue; ++id; int v = u; for (; v != -1 && state[v] == 0; v = pred[v] == INVALID ? -1 : rgr.id(rgr.source(pred[v]))) { state[v] = id; } if (v != -1 && state[v] == id) { // A negative cycle is found cycle_found = true; cycle.clear(); StaticDigraph::Arc a = pred[v]; Value d, delta = _res_cap[rgr.id(a)]; cycle.push_back(rgr.id(a)); while (rgr.id(rgr.source(a)) != v) { a = pred_map[rgr.source(a)]; d = _res_cap[rgr.id(a)]; if (d < delta) delta = d; cycle.push_back(rgr.id(a)); } // Augment along the cycle for (int i = 0; i < int(cycle.size()); ++i) { int j = cycle[i]; _res_cap[j] -= delta; _res_cap[_reverse[j]] += delta; } } } } // Increase iteration limit if no cycle is found if (!cycle_found) { length_bound = static_cast(length_bound * BF_LIMIT_FACTOR); } } } } // Execute the "Minimum Mean Cycle Canceling" method void startMinMeanCycleCanceling() { typedef Path SPath; typedef typename SPath::ArcIt SPathArcIt; typedef typename HowardMmc ::template SetPath::Create HwMmc; typedef typename HartmannOrlinMmc ::template SetPath::Create HoMmc; const double HW_ITER_LIMIT_FACTOR = 1.0; const int HW_ITER_LIMIT_MIN_VALUE = 5; const int hw_iter_limit = std::max(static_cast(HW_ITER_LIMIT_FACTOR * _node_num), HW_ITER_LIMIT_MIN_VALUE); SPath cycle; HwMmc hw_mmc(_sgr, _cost_map); hw_mmc.cycle(cycle); buildResidualNetwork(); while (true) { typename HwMmc::TerminationCause hw_tc = hw_mmc.findCycleMean(hw_iter_limit); if (hw_tc == HwMmc::ITERATION_LIMIT) { // Howard's algorithm reached the iteration limit, start a // strongly polynomial algorithm instead HoMmc ho_mmc(_sgr, _cost_map); ho_mmc.cycle(cycle); // Find a minimum mean cycle (Hartmann-Orlin algorithm) if (!(ho_mmc.findCycleMean() && ho_mmc.cycleCost() < 0)) break; ho_mmc.findCycle(); } else { // Find a minimum mean cycle (Howard algorithm) if (!(hw_tc == HwMmc::OPTIMAL && hw_mmc.cycleCost() < 0)) break; hw_mmc.findCycle(); } // Compute delta value Value delta = INF; for (SPathArcIt a(cycle); a != INVALID; ++a) { Value d = _res_cap[_id_vec[_sgr.id(a)]]; if (d < delta) delta = d; } // Augment along the cycle for (SPathArcIt a(cycle); a != INVALID; ++a) { int j = _id_vec[_sgr.id(a)]; _res_cap[j] -= delta; _res_cap[_reverse[j]] += delta; } // Rebuild the residual network buildResidualNetwork(); } } // Execute the "Cancel-and-Tighten" method void startCancelAndTighten() { // Constants for the min mean cycle computations const double LIMIT_FACTOR = 1.0; const int MIN_LIMIT = 5; const double HW_ITER_LIMIT_FACTOR = 1.0; const int HW_ITER_LIMIT_MIN_VALUE = 5; const int hw_iter_limit = std::max(static_cast(HW_ITER_LIMIT_FACTOR * _node_num), HW_ITER_LIMIT_MIN_VALUE); // Contruct auxiliary data vectors DoubleVector pi(_res_node_num, 0.0); IntVector level(_res_node_num); BoolVector reached(_res_node_num); BoolVector processed(_res_node_num); IntVector pred_node(_res_node_num); IntVector pred_arc(_res_node_num); std::vector stack(_res_node_num); std::vector proc_vector(_res_node_num); // Initialize epsilon double epsilon = 0; for (int a = 0; a != _res_arc_num; ++a) { if (_res_cap[a] > 0 && -_cost[a] > epsilon) epsilon = -_cost[a]; } // Start phases Tolerance tol; tol.epsilon(1e-6); int limit = int(LIMIT_FACTOR * std::sqrt(double(_res_node_num))); if (limit < MIN_LIMIT) limit = MIN_LIMIT; int iter = limit; while (epsilon * _res_node_num >= 1) { // Find and cancel cycles in the admissible network using DFS for (int u = 0; u != _res_node_num; ++u) { reached[u] = false; processed[u] = false; } int stack_head = -1; int proc_head = -1; for (int start = 0; start != _res_node_num; ++start) { if (reached[start]) continue; // New start node reached[start] = true; pred_arc[start] = -1; pred_node[start] = -1; // Find the first admissible outgoing arc double p = pi[start]; int a = _first_out[start]; int last_out = _first_out[start+1]; for (; a != last_out && (_res_cap[a] == 0 || !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ; if (a == last_out) { processed[start] = true; proc_vector[++proc_head] = start; continue; } stack[++stack_head] = a; while (stack_head >= 0) { int sa = stack[stack_head]; int u = _source[sa]; int v = _target[sa]; if (!reached[v]) { // A new node is reached reached[v] = true; pred_node[v] = u; pred_arc[v] = sa; p = pi[v]; a = _first_out[v]; last_out = _first_out[v+1]; for (; a != last_out && (_res_cap[a] == 0 || !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ; stack[++stack_head] = a == last_out ? -1 : a; } else { if (!processed[v]) { // A cycle is found int n, w = u; Value d, delta = _res_cap[sa]; for (n = u; n != v; n = pred_node[n]) { d = _res_cap[pred_arc[n]]; if (d <= delta) { delta = d; w = pred_node[n]; } } // Augment along the cycle _res_cap[sa] -= delta; _res_cap[_reverse[sa]] += delta; for (n = u; n != v; n = pred_node[n]) { int pa = pred_arc[n]; _res_cap[pa] -= delta; _res_cap[_reverse[pa]] += 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 outgoing arc p = pi[v]; a = stack[stack_head] + 1; last_out = _first_out[v+1]; for (; a != last_out && (_res_cap[a] == 0 || !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ; stack[stack_head] = a == last_out ? -1 : a; } while (stack_head >= 0 && stack[stack_head] == -1) { processed[v] = true; proc_vector[++proc_head] = v; if (--stack_head >= 0) { // Find the next admissible outgoing arc v = _source[stack[stack_head]]; p = pi[v]; a = stack[stack_head] + 1; last_out = _first_out[v+1]; for (; a != last_out && (_res_cap[a] == 0 || !tol.negative(_cost[a] + p - pi[_target[a]])); ++a) ; stack[stack_head] = a == last_out ? -1 : a; } } } } // Tighten potentials and epsilon if (--iter > 0) { for (int u = 0; u != _res_node_num; ++u) { level[u] = 0; } for (int i = proc_head; i > 0; --i) { int u = proc_vector[i]; double p = pi[u]; int l = level[u] + 1; int last_out = _first_out[u+1]; for (int a = _first_out[u]; a != last_out; ++a) { int v = _target[a]; if (_res_cap[a] > 0 && tol.negative(_cost[a] + p - pi[v]) && l > level[v]) level[v] = l; } } // Modify potentials double q = std::numeric_limits::max(); for (int u = 0; u != _res_node_num; ++u) { int lu = level[u]; double p, pu = pi[u]; int last_out = _first_out[u+1]; for (int a = _first_out[u]; a != last_out; ++a) { if (_res_cap[a] == 0) continue; int v = _target[a]; int ld = lu - level[v]; if (ld > 0) { p = (_cost[a] + pu - pi[v] + epsilon) / (ld + 1); if (p < q) q = p; } } } for (int u = 0; u != _res_node_num; ++u) { pi[u] -= q * level[u]; } // Modify epsilon epsilon = 0; for (int u = 0; u != _res_node_num; ++u) { double curr, pu = pi[u]; int last_out = _first_out[u+1]; for (int a = _first_out[u]; a != last_out; ++a) { if (_res_cap[a] == 0) continue; curr = _cost[a] + pu - pi[_target[a]]; if (-curr > epsilon) epsilon = -curr; } } } else { typedef HowardMmc HwMmc; typedef HartmannOrlinMmc HoMmc; typedef typename BellmanFord ::template SetDistMap::Create BF; // Set epsilon to the minimum cycle mean Cost cycle_cost = 0; int cycle_size = 1; buildResidualNetwork(); HwMmc hw_mmc(_sgr, _cost_map); if (hw_mmc.findCycleMean(hw_iter_limit) == HwMmc::ITERATION_LIMIT) { // Howard's algorithm reached the iteration limit, start a // strongly polynomial algorithm instead HoMmc ho_mmc(_sgr, _cost_map); ho_mmc.findCycleMean(); epsilon = -ho_mmc.cycleMean(); cycle_cost = ho_mmc.cycleCost(); cycle_size = ho_mmc.cycleSize(); } else { // Set epsilon epsilon = -hw_mmc.cycleMean(); cycle_cost = hw_mmc.cycleCost(); cycle_size = hw_mmc.cycleSize(); } // Compute feasible potentials for the current epsilon for (int i = 0; i != int(_cost_vec.size()); ++i) { _cost_vec[i] = cycle_size * _cost_vec[i] - cycle_cost; } BF bf(_sgr, _cost_map); bf.distMap(_pi_map); bf.init(0); bf.start(); for (int u = 0; u != _res_node_num; ++u) { pi[u] = static_cast(_pi[u]) / cycle_size; } iter = limit; } } } }; //class CycleCanceling ///@} } //namespace lemon #endif //LEMON_CYCLE_CANCELING_H