NetworkSimplex< GR, V, C > Class Template Reference

Detailed Description

template<typename GR, typename V = int, typename C = V>
class lemon::NetworkSimplex< GR, V, C >

NetworkSimplex implements the primal Network Simplex algorithm for finding a 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 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 run() function. If some parameters are not specified, then default values will be used.

Template Parameters

GR	The digraph type the algorithm runs on.
V	The value type used for flow amounts, capacity bounds and supply values in the algorithm. By default it is int.
C	The value type used for costs and potentials in the algorithm. By default it is the same as 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 PivotRule.

#include <lemon/network_simplex.h>

Public Types
enum	ProblemType { INFEASIBLE, OPTIMAL, UNBOUNDED }
	Problem type constants for the run() function. More...
enum	SupplyType { GEQ, LEQ }
	Constants for selecting the type of the supply constraints. More...
enum	PivotRule {   FIRST_ELIGIBLE, BEST_ELIGIBLE, BLOCK_SEARCH, CANDIDATE_LIST,   ALTERING_LIST }
	Constants for selecting the pivot rule. More...
typedef V	Value
	The type of the flow amounts, capacity bounds and supply values.
typedef C	Cost
	The type of the arc costs.

Public Member Functions
	NetworkSimplex (const GR &graph)
	Constructor.
Parameters
The parameters of the algorithm can be specified using these functions.
template<typename LowerMap >
NetworkSimplex &	lowerMap (const LowerMap &map)
	Set the lower bounds on the arcs.
template<typename UpperMap >
NetworkSimplex &	upperMap (const UpperMap &map)
	Set the upper bounds (capacities) on the arcs.
template<typename CostMap >
NetworkSimplex &	costMap (const CostMap &map)
	Set the costs of the arcs.
template<typename SupplyMap >
NetworkSimplex &	supplyMap (const SupplyMap &map)
	Set the supply values of the nodes.
NetworkSimplex &	stSupply (const Node &s, const Node &t, Value k)
	Set single source and target nodes and a supply value.
NetworkSimplex &	supplyType (SupplyType supply_type)
	Set the type of the supply constraints.
Execution Control
The algorithm can be executed using run().
ProblemType	run (PivotRule pivot_rule=BLOCK_SEARCH)
	Run the algorithm.
NetworkSimplex &	reset ()
	Reset all the parameters that have been given before.
Query Functions
The results of the algorithm can be obtained using these functions. The run() function must be called before using them.
template<typename Number >
Number	totalCost () const
	Return the total cost of the found flow.
Value	flow (const Arc &a) const
	Return the flow on the given arc.
template<typename FlowMap >
void	flowMap (FlowMap &map) const
	Return the flow map (the primal solution).
Cost	potential (const Node &n) const
	Return the potential (dual value) of the given node.
template<typename PotentialMap >
void	potentialMap (PotentialMap &map) const
	Return the potential map (the dual solution).

Public Attributes
const Value	INF
	Constant for infinite upper bounds (capacities).

Member Enumeration Documentation

enum ProblemType

Enum type containing the problem type constants that can be returned by the run() function of the algorithm.

Enumerator:

INFEASIBLE	The problem has no feasible solution (flow).
OPTIMAL	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).
UNBOUNDED	The objective function of the problem is unbounded, i.e. there is a directed cycle having negative total cost and infinite upper bound.

enum SupplyType

Enum type containing constants for selecting the supply type, i.e. the direction of the inequalities in the supply/demand constraints of the minimum cost flow problem.

The default supply type is GEQ, the LEQ type can be selected using supplyType(). The equality form is a special case of both supply types.

Enumerator:

GEQ	This option means that there are "greater or equal" supply/demand constraints in the definition of the problem.
LEQ	This option means that there are "less or equal" supply/demand constraints in the definition of the problem.

enum PivotRule

Enum type containing constants for selecting the pivot rule for the run() function.

NetworkSimplex provides five different pivot rule implementations that significantly affect the running time of the algorithm. By default 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 run() function with the proper parameter.

Enumerator:

FIRST_ELIGIBLE	The First Eligible pivot rule. The next eligible arc is selected in a wraparound fashion in every iteration.
BEST_ELIGIBLE	The Best Eligible pivot rule. The best eligible arc is selected in every iteration.
BLOCK_SEARCH	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.
CANDIDATE_LIST	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.
ALTERING_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.

Constructor & Destructor Documentation

NetworkSimplex ( const GR &  graph )

inline

The constructor of the class.

Parameters

graph

The digraph the algorithm runs on.

Member Function Documentation

NetworkSimplex& lowerMap ( const LowerMap &  map )

inline

This function sets the lower bounds on the arcs. If it is not used before calling run(), the lower bounds will be set to zero on all arcs.

Parameters

map	An arc map storing the lower bounds. Its Value type must be convertible to the Value type of the algorithm.

Returns

(*this)

NetworkSimplex& upperMap ( const UpperMap &  map )

inline

This function sets the upper bounds (capacities) on the arcs. If it is not used before calling run(), the upper bounds will be set to INF on all arcs (i.e. the flow value will be unbounded from above on each arc).

Parameters

map	An arc map storing the upper bounds. Its Value type must be convertible to the Value type of the algorithm.

Returns

(*this)

NetworkSimplex& costMap ( const CostMap &  map )

inline

This function sets the costs of the arcs. If it is not used before calling run(), the costs will be set to 1 on all arcs.

Parameters

map	An arc map storing the costs. Its Value type must be convertible to the Cost type of the algorithm.

Returns

(*this)

NetworkSimplex& supplyMap ( const SupplyMap &  map )

inline

This function sets the supply values of the nodes. If neither this function nor stSupply() is used before calling run(), the supply of each node will be set to zero. (It makes sense only if non-zero lower bounds are given.)

Parameters

map	A node map storing the supply values. Its Value type must be convertible to the Value type of the algorithm.

Returns

(*this)

NetworkSimplex& stSupply ( const Node &  s, const Node &  t, Value  k  )

inline

This function sets a single source node and a single target node and the required flow value. If neither this function nor supplyMap() is used before calling 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 supplyMap() with such a map in which k is assigned to s, -k is assigned to t and all other nodes have zero supply value.

Parameters

s	The source node.
t	The target node.
k	The required amount of flow from node s to node t (i.e. the supply of s and the demand of t).

Returns

(*this)

NetworkSimplex& supplyType ( SupplyType  supply_type )

inline

This function sets the type of the supply/demand constraints. If it is not used before calling run(), the GEQ supply type will be used.

For more information see SupplyType.

Returns

(*this)

ProblemType run ( PivotRule  pivot_rule = BLOCK_SEARCH )

inline

This function runs the algorithm. The paramters can be specified using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(), supplyType(). For example,

NetworkSimplex<ListDigraph> ns(graph);
ns.lowerMap(lower).upperMap(upper).costMap(cost)
  .supplyMap(sup).run();

This function can be called more than once. All the parameters that have been given are kept for the next call, unless reset() is called, thus only the modified parameters have to be set again. See reset() for examples. However the underlying digraph must not be modified after this class have been constructed, since it copies and extends the graph.

Parameters

pivot_rule

The pivot rule that will be used during the algorithm. For more information see PivotRule.

Returns

INFEASIBLE if no feasible flow exists,
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),
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 Also

ProblemType, PivotRule

NetworkSimplex& reset ( )

inline

This function resets all the paramaters that have been given before using functions lowerMap(), upperMap(), costMap(), supplyMap(), stSupply(), supplyType().

It is useful for multiple run() calls. If this function is not used, all the parameters given before are kept for the next 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,

NetworkSimplex<ListDigraph> 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();

Returns

(*this)

Number totalCost ( ) const

inline

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,

ns.totalCost<double>();

It is useful if the total cost cannot be stored in the Cost type of the algorithm, which is the default return type of the function.

Precondition

run() must be called before using this function.

Value flow ( const Arc &  a ) const

inline

This function returns the flow on the given arc.

Precondition

run() must be called before using this function.

void flowMap ( FlowMap &  map ) const

inline

This function copies the flow value on each arc into the given map. The Value type of the algorithm must be convertible to the Value type of the map.

Precondition

run() must be called before using this function.

Cost potential ( const Node &  n ) const

inline

This function returns the potential (dual value) of the given node.

Precondition

run() must be called before using this function.

void potentialMap ( PotentialMap &  map ) const

inline

This function copies the potential (dual value) of each node into the given map. The Cost type of the algorithm must be convertible to the Value type of the map.

Precondition

run() must be called before using this function.

Member Data Documentation

const Value INF

Constant for infinite upper bounds (capacities). It is std::numeric_limits<Value>::infinity() if available, std::numeric_limits<Value>::max() otherwise.