MaxFractionalMatching< GR, TR > Class Template Reference

Detailed Description

template<typename GR, typename TR>
class lemon::MaxFractionalMatching< GR, TR >

This class provides an implementation of fractional matching algorithm based on push-relabel principle.

The maximum cardinality fractional matching is a relaxation of the maximum cardinality matching problem where the odd set constraints are omitted. It can be formulated with the following linear program.

\[ \sum_{e \in \delta(u)}x_e \le 1 \quad \forall u\in V\]

\[x_e \ge 0\quad \forall e\in E\]

\[\max \sum_{e\in E}x_e\]

where \(\delta(X)\) is the set of edges incident to a node in \(X\). The result can be represented as the union of a matching with one value edges and a set of odd length cycles with half value edges.

The algorithm calculates an optimal fractional matching and a barrier. The number of adjacents of any node set minus the size of node set is a lower bound on the uncovered nodes in the graph. For maximum matching a barrier is computed which maximizes this difference.

The algorithm can be executed with the run() function. After it the matching (the primal solution) and the barrier (the dual solution) can be obtained using the query functions.

The primal solution is multiplied by 2.

Template Parameters

GR	The undirected graph type the algorithm runs on.

#include <lemon/fractional_matching.h>

Classes
struct	SetElevator
	Named parameter for setting Elevator type More...
struct	SetMatchingMap
	Named parameter for setting MatchingMap type More...
struct	SetStandardElevator
	Named parameter for setting Elevator type with automatic allocation More...

Public Types
typedef TR	Traits
	The traits class of the algorithm.
typedef TR::Graph	Graph
	The type of the graph the algorithm runs on.
typedef TR::MatchingMap	MatchingMap
	The type of the matching map.
typedef TR::Elevator	Elevator
	The type of the elevator.

Public Member Functions
	MaxFractionalMatching (const Graph &graph, bool allow_loops=true)
MaxFractionalMatching &	matchingMap (MatchingMap &map)
	Sets the matching map.
MaxFractionalMatching &	elevator (Elevator &elevator)
	Sets the elevator used by algorithm.
const Elevator &	elevator () const
	Returns a const reference to the elevator.
Execution control
The simplest way to execute the algorithm is to use one of the member functions called run(). If you need more control on the execution, first you must call init() and then one variant of the start() member.
void	init ()
	Initializes the internal data structures.
void	start (bool postprocess=true)
	Starts the algorithm and computes a fractional matching.
bool	startPerfect (bool postprocess=true)
	Starts the algorithm and computes a perfect fractional matching.
void	run (bool postprocess=true)
	Runs the algorithm.
bool	runPerfect (bool postprocess=true)
	Runs the algorithm to find a perfect fractional matching.
Query Functions
The result of the Matching algorithm can be obtained using these functions. Before the use of these functions, either run() or start() must be called.
int	matchingSize () const
	Return the number of covered nodes in the matching.
const MatchingMap &	matchingMap () const
	Returns a const reference to the matching map.
int	matching (const Edge &edge) const
	Return true if the given edge is in the matching.
Arc	matching (const Node &node) const
	Return the fractional matching arc (or edge) incident to the given node.
bool	barrier (const Node &node) const
	Returns true if the node is in the barrier.

Static Public Attributes
static const int	primalScale = 2

Constructor & Destructor Documentation

MaxFractionalMatching ( const Graph &  graph, bool  allow_loops = true  )

inline

Constructor.

Member Function Documentation

MaxFractionalMatching& matchingMap ( MatchingMap &  map )

inline

Sets the matching map. If you don't use this function before calling run() or init(), an instance will be allocated automatically. The destructor deallocates this automatically allocated map, of course.

Returns

(*this)

MaxFractionalMatching& elevator ( Elevator &  elevator )

inline

Sets the elevator used by algorithm. If you don't use this function before calling run() or init(), an instance will be allocated automatically. The destructor deallocates this automatically allocated elevator, of course.

Returns

(*this)

const Elevator& elevator ( ) const

inline

Returns a const reference to the elevator.

Precondition

Either run() or init() must be called before using this function.

void init ( )

inline

Initializes the internal data structures and sets the initial matching.

void start ( bool  postprocess = true )

inline

The algorithm computes a maximum fractional matching.

Parameters

postprocess

The algorithm computes first a matching which is a union of a matching with one value edges, cycles with half value edges and even length paths with half value edges. If the parameter is true, then after the push-relabel algorithm it postprocesses the matching to contain only matching edges and half value odd cycles.

bool startPerfect ( bool  postprocess = true )

inline

The algorithm computes a perfect fractional matching. If it does not exists, then the algorithm returns false and the matching is undefined and the barrier.

Parameters

postprocess

The algorithm computes first a matching which is a union of a matching with one value edges, cycles with half value edges and even length paths with half value edges. If the parameter is true, then after the push-relabel algorithm it postprocesses the matching to contain only matching edges and half value odd cycles.

void run ( bool  postprocess = true )

inline

Just a shortcut for the next code:

init();
start();

bool runPerfect ( bool  postprocess = true )

inline

Just a shortcut for the next code:

init();
startPerfect();

int matchingSize ( ) const

inline

This function returns the number of covered nodes in the matching.

Precondition

Either run() or start() must be called before using this function.

const MatchingMap& matchingMap ( ) const

inline

Returns a const reference to the node map storing the found fractional matching. This method can be called after running the algorithm.

Precondition

Either run() or init() must be called before using this function.

int matching ( const Edge &  edge ) const

inline

This function returns true if the given edge is in the found matching. The result is scaled by primal scale.

Precondition

Either run() or start() must be called before using this function.

Arc matching ( const Node &  node ) const

inline

This function returns one of the fractional matching arc (or edge) incident to the given node in the found matching or INVALID if the node is not covered by the matching or if the node is on an odd length cycle then it is the successor edge on the cycle.

Precondition

Either run() or start() must be called before using this function.

bool barrier ( const Node &  node ) const

inline

The barrier is a subset of the nodes. If the nodes in the barrier have less adjacent nodes than the size of the barrier, then at least as much nodes cannot be covered as the difference of the two subsets.

Member Data Documentation

const int primalScale = 2

static

Scaling factor for primal solution.