/* -*- mode: C++; indent-tabs-mode: nil; -*-
* This file is a part of LEMON, a generic C++ optimization library.
* Copyright (C) 2003-2010
* 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
#ifndef LEMON_CAPACITY_SCALING_H
#define LEMON_CAPACITY_SCALING_H
/// \ingroup min_cost_flow_algs
/// \brief Capacity Scaling algorithm for finding a minimum cost flow.
#include <lemon/bin_heap.h>
/// \brief Default traits class of CapacityScaling algorithm.
/// Default traits class of CapacityScaling algorithm.
/// \tparam GR Digraph type.
/// \tparam V The number type used for flow amounts, capacity bounds
/// and supply values. By default it is \c int.
/// \tparam C The number type used for costs and potentials.
/// By default it is the same as \c V.
template <typename GR, typename V = int, typename C = V>
struct CapacityScalingDefaultTraits
/// The type of the digraph
/// The type of the flow amounts, capacity bounds and supply values
/// The type of the arc costs
/// \brief The type of the heap used for internal Dijkstra computations.
/// The type of the heap used for internal Dijkstra computations.
/// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
/// its priority type must be \c Cost and its cross reference type
/// must be \ref RangeMap "RangeMap<int>".
typedef BinHeap<Cost, RangeMap<int> > Heap;
/// \addtogroup min_cost_flow_algs
/// \brief Implementation of the Capacity Scaling algorithm for
/// finding a \ref min_cost_flow "minimum cost flow".
/// \ref CapacityScaling implements the capacity scaling version
/// of the successive shortest path algorithm for finding a
/// \ref min_cost_flow "minimum cost flow" \ref amo93networkflows,
/// \ref edmondskarp72theoretical. It is an efficient dual
/// 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.
/// \tparam TR The traits class that defines various types used by the
/// algorithm. By default, it is \ref CapacityScalingDefaultTraits
/// "CapacityScalingDefaultTraits<GR, V, C>".
/// In most cases, this parameter should not be set directly,
/// consider to use the named template parameters instead.
/// \warning Both number types must be signed and all input data must
/// \warning This algorithm does not support negative costs for such
/// arcs that have infinite upper bound.
template <typename GR, typename V, typename C, typename TR>
template < typename GR, typename V = int, typename C = V,
typename TR = CapacityScalingDefaultTraits<GR, V, C> >
/// The type of the digraph
typedef typename TR::Digraph Digraph;
/// The type of the flow amounts, capacity bounds and supply values
typedef typename TR::Value Value;
/// The type of the arc costs
typedef typename TR::Cost Cost;
/// The type of the heap used for internal Dijkstra computations
typedef typename TR::Heap Heap;
/// The \ref CapacityScalingDefaultTraits "traits class" of the algorithm
/// \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.
/// The problem has no feasible solution (flow).
/// 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).
/// 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
TEMPLATE_DIGRAPH_TYPEDEFS(GR);
typedef std::vector<int> IntVector;
typedef std::vector<Value> ValueVector;
typedef std::vector<Cost> CostVector;
typedef std::vector<char> BoolVector;
// Note: vector<char> is used instead of vector<bool> for efficiency reasons
// Data related to the underlying digraph
// Parameters of the problem
// Data structures for storing the digraph
IntVector _deficit_nodes;
/// \brief Constant for infinite upper bounds (capacities).
/// Constant for infinite upper bounds (capacities).
/// It is \c std::numeric_limits<Value>::infinity() if available,
/// \c std::numeric_limits<Value>::max() otherwise.
// Special implementation of the Dijkstra algorithm for finding
// shortest paths in the residual network of the digraph with
// respect to the reduced arc costs and modifying the node
// potentials according to the found distance labels.
const IntVector &_first_out;
const IntVector &_target;
const ValueVector &_res_cap;
const ValueVector &_excess;
ResidualDijkstra(CapacityScaling& cs) :
_node_num(cs._node_num), _geq(cs._sum_supply < 0),
_first_out(cs._first_out), _target(cs._target), _cost(cs._cost),
_res_cap(cs._res_cap), _excess(cs._excess), _pi(cs._pi),
_pred(cs._pred), _dist(cs._node_num)
int run(int s, Value delta = 1) {
RangeMap<int> heap_cross_ref(_node_num, Heap::PRE_HEAP);
Heap heap(heap_cross_ref);
while (!heap.empty() && _excess[heap.top()] > -delta) {
Cost d = heap.prio() + _pi[u], dn;
_proc_nodes.push_back(u);
// Traverse outgoing residual arcs
int last_out = _geq ? _first_out[u+1] : _first_out[u+1] - 1;
for (int a = _first_out[u]; a != last_out; ++a) {
if (_res_cap[a] < delta) continue;
heap.push(v, d + _cost[a] - _pi[v]);
dn = d + _cost[a] - _pi[v];
if (heap.empty()) return -1;
// Update potentials of processed nodes
for (int i = 0; i < int(_proc_nodes.size()); ++i) {
_pi[_proc_nodes[i]] += _dist[_proc_nodes[i]] - dt;
}; //class ResidualDijkstra
/// \name Named Template Parameters
struct SetHeapTraits : public Traits {
/// \brief \ref named-templ-param "Named parameter" for setting
/// \ref named-templ-param "Named parameter" for setting \c Heap
/// type, which is used for internal Dijkstra computations.
/// It must conform to the \ref lemon::concepts::Heap "Heap" concept,
/// its priority type must be \c Cost and its cross reference type
/// must be \ref RangeMap "RangeMap<int>".
: public CapacityScaling<GR, V, C, SetHeapTraits<T> > {
typedef CapacityScaling<GR, V, C, SetHeapTraits<T> > Create;
/// The constructor of the class.
/// \param graph The digraph the algorithm runs on.
CapacityScaling(const GR& graph) :
_graph(graph), _node_id(graph), _arc_idf(graph), _arc_idb(graph),
INF(std::numeric_limits<Value>::has_infinity ?
std::numeric_limits<Value>::infinity() :
std::numeric_limits<Value>::max())
// Check the number types
LEMON_ASSERT(std::numeric_limits<Value>::is_signed,
"The flow type of CapacityScaling must be signed");
LEMON_ASSERT(std::numeric_limits<Cost>::is_signed,
"The cost type of CapacityScaling must be signed");
/// The parameters of the algorithm can be specified using these
/// \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
/// \return <tt>(*this)</tt>
template <typename LowerMap>
CapacityScaling& lowerMap(const LowerMap& map) {
for (ArcIt a(_graph); a != INVALID; ++a) {
_lower[_arc_idf[a]] = map[a];
_lower[_arc_idb[a]] = map[a];
/// \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
/// \return <tt>(*this)</tt>
template<typename UpperMap>
CapacityScaling& upperMap(const UpperMap& map) {
for (ArcIt a(_graph); a != INVALID; ++a) {
_upper[_arc_idf[a]] = map[a];
/// \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
/// \return <tt>(*this)</tt>
template<typename CostMap>
CapacityScaling& costMap(const CostMap& map) {
for (ArcIt a(_graph); a != INVALID; ++a) {
_cost[_arc_idf[a]] = map[a];
_cost[_arc_idb[a]] = -map[a];
/// \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
/// \return <tt>(*this)</tt>
template<typename SupplyMap>
CapacityScaling& supplyMap(const SupplyMap& map) {
for (NodeIt n(_graph); n != INVALID; ++n) {
_supply[_node_id[n]] = map[n];
/// \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 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 <tt>(*this)</tt>
CapacityScaling& stSupply(const Node& s, const Node& t, Value k) {
for (int i = 0; i != _node_num; ++i) {
_supply[_node_id[s]] = k;
_supply[_node_id[t]] = -k;
/// \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().
/// CapacityScaling<ListDigraph> cs(graph);
/// cs.lowerMap(lower).upperMap(upper).costMap(cost)
/// .supplyMap(sup).run();
/// 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 factor The capacity scaling factor. It must be larger than
/// one to use scaling. If it is less or equal to one, then scaling
/// \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
/// \see resetParams(), reset()
ProblemType run(int factor = 4) {
if (pt != OPTIMAL) return pt;
/// \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.
/// CapacityScaling<ListDigraph> cs(graph);
/// cs.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)
/// cs.costMap(cost).run();
/// // Run again from scratch using resetParams()
/// // (the lower bounds will be set to zero on all arcs)
/// cs.upperMap(capacity).costMap(cost)
/// .supplyMap(sup).run();
/// \return <tt>(*this)</tt>
CapacityScaling& resetParams() {
for (int i = 0; i != _node_num; ++i) {
for (int j = 0; j != _res_arc_num; ++j) {
_cost[j] = _forward[j] ? 1 : -1;
/// \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 <tt>(*this)</tt>
/// \see resetParams(), run()
CapacityScaling& reset() {
_node_num = countNodes(_graph);
_arc_num = countArcs(_graph);
_res_arc_num = 2 * (_arc_num + _node_num);
_first_out.resize(_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(_node_num);
_res_cap.resize(_res_arc_num);
_excess.resize(_node_num);
int i = 0, j = 0, k = 2 * _arc_num + _node_num - 1;
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
for (NodeIt n(_graph); n != INVALID; ++n, ++i) {
for (OutArcIt a(_graph, n); a != INVALID; ++a, ++j) {
_target[j] = _node_id[_graph.runningNode(a)];
for (InArcIt a(_graph, n); a != INVALID; ++a, ++j) {
_target[j] = _node_id[_graph.runningNode(a)];
_first_out[_node_num] = k;
for (ArcIt a(_graph); a != INVALID; ++a) {
/// \name Query Functions
/// The results of the algorithm can be obtained using these
/// 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,
/// cs.totalCost<double>();
/// 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
/// \pre \ref run() must be called before using this function.
template <typename Number>
Number totalCost() const {
for (ArcIt a(_graph); a != INVALID; ++a) {
c += static_cast<Number>(_res_cap[i]) *
(-static_cast<Number>(_cost[i]));
return totalCost<Cost>();
/// \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 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 <typename FlowMap>
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
/// \pre \ref run() must be called before using this function.
Cost potential(const Node& n) const {
/// \brief Return the potential map (the dual solution).
/// This function copies the potential (dual value) of each node
/// 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 <typename PotentialMap>
void potentialMap(PotentialMap &map) const {
for (NodeIt n(_graph); n != INVALID; ++n) {
map.set(n, _pi[_node_id[n]]);
// Initialize the algorithm
if (_node_num <= 1) return INFEASIBLE;
// Check the sum of supply values
for (int i = 0; i != _root; ++i) {
_sum_supply += _supply[i];
if (_sum_supply > 0) return INFEASIBLE;
for (int i = 0; i != _root; ++i) {
// Remove non-zero lower bounds
const Value MAX = std::numeric_limits<Value>::max();
for (int i = 0; i != _root; ++i) {
last_out = _first_out[i+1];
for (int j = _first_out[i]; j != last_out; ++j) {
_res_cap[j] = _upper[j] < MAX ? _upper[j] - c : INF;
_res_cap[j] = _upper[j] < MAX + c ? _upper[j] - c : INF;
_excess[_target[j]] += c;
for (int j = 0; j != _res_arc_num; ++j) {
_res_cap[j] = _forward[j] ? _upper[j] : 0;
for (int i = 0; i != _root; ++i) {
last_out = _first_out[i+1] - 1;
for (int j = _first_out[i]; j != last_out; ++j) {
if (_cost[j] < 0 && rc > 0) {
if (rc >= MAX) return UNBOUNDED;
_excess[_target[j]] += rc;
_res_cap[_reverse[j]] += rc;
// Handle GEQ supply type
_excess[_root] = -_sum_supply;
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
_res_cap[a] = -_sum_supply + 1;
for (int a = _first_out[_root]; a != _res_arc_num; ++a) {
// Initialize delta value
Value max_sup = 0, max_dem = 0, max_cap = 0;
for (int i = 0; i != _root; ++i) {
if ( ex > max_sup) max_sup = ex;
if (-ex > max_dem) max_dem = -ex;
int last_out = _first_out[i+1] - 1;
for (int j = _first_out[i]; j != last_out; ++j) {
if (_res_cap[j] > max_cap) max_cap = _res_cap[j];
max_sup = std::min(std::min(max_sup, max_dem), max_cap);
for (_delta = 1; 2 * _delta <= max_sup; _delta *= 2) ;
pt = startWithoutScaling();
// Handle non-zero lower bounds
int limit = _first_out[_root];
for (int j = 0; j != limit; ++j) {
if (!_forward[j]) _res_cap[j] += _lower[j];
// Shift potentials if necessary
if (_sum_supply < 0 || pr > 0) {
for (int i = 0; i != _node_num; ++i) {
// Execute the capacity scaling algorithm
ProblemType startWithScaling() {
// Perform capacity scaling phases
ResidualDijkstra _dijkstra(*this);
// Saturate all arcs not satisfying the optimality condition
for (int u = 0; u != _node_num; ++u) {
last_out = _sum_supply < 0 ?
_first_out[u+1] : _first_out[u+1] - 1;
for (int a = _first_out[u]; a != last_out; ++a) {
Cost c = _cost[a] + _pi[u] - _pi[v];
if (c < 0 && rc >= _delta) {
_res_cap[_reverse[a]] += rc;
// Find excess nodes and deficit nodes
for (int u = 0; u != _node_num; ++u) {
if (ex >= _delta) _excess_nodes.push_back(u);
if (ex <= -_delta) _deficit_nodes.push_back(u);
int next_node = 0, next_def_node = 0;
// Find augmenting shortest paths
while (next_node < int(_excess_nodes.size())) {
bool delta_deficit = false;
for ( ; next_def_node < int(_deficit_nodes.size());
if (_excess[_deficit_nodes[next_def_node]] <= -_delta) {
if (!delta_deficit) break;
// Run Dijkstra in the residual network
s = _excess_nodes[next_node];
if ((t = _dijkstra.run(s, _delta)) == -1) {
// Augment along a shortest path from s to t
Value d = std::min(_excess[s], -_excess[t]);
while ((a = _pred[u]) != -1) {
if (_res_cap[a] < d) d = _res_cap[a];
while ((a = _pred[u]) != -1) {
_res_cap[_reverse[a]] += d;
if (_excess[s] < _delta) ++next_node;
_delta = _delta <= _factor ? 1 : _delta / _factor;
// Execute the successive shortest path algorithm
ProblemType startWithoutScaling() {
for (int i = 0; i != _node_num; ++i) {
if (_excess[i] > 0) _excess_nodes.push_back(i);
if (_excess_nodes.size() == 0) return OPTIMAL;
ResidualDijkstra _dijkstra(*this);
while ( _excess[_excess_nodes[next_node]] > 0 ||
++next_node < int(_excess_nodes.size()) )
// Run Dijkstra in the residual network
s = _excess_nodes[next_node];
if ((t = _dijkstra.run(s)) == -1) return INFEASIBLE;
// Augment along a shortest path from s to t
Value d = std::min(_excess[s], -_excess[t]);
while ((a = _pred[u]) != -1) {
if (_res_cap[a] < d) d = _res_cap[a];
while ((a = _pred[u]) != -1) {
_res_cap[_reverse[a]] += d;
}; //class CapacityScaling
#endif //LEMON_CAPACITY_SCALING_H