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MaxWeightedFractionalMatching< GR, WM > Class Template Reference

Detailed Description

template<typename GR, typename WM>
class lemon::MaxWeightedFractionalMatching< GR, WM >

This class provides an efficient implementation of fractional matching algorithm. The implementation uses priority queues and provides \(O(nm\log n)\) time complexity.

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

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

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

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

where \(\delta(X)\) is the set of edges incident to a node in \(X\). The result must be 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 proof of the optimality. The solution of the dual problem can be used to check the result of the algorithm. The dual linear problem is the following.

\[ y_u + y_v \ge w_{uv} \quad \forall uv\in E\]

\[y_u \ge 0 \quad \forall u \in V\]

\[\min \sum_{u \in V}y_u \]

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

The primal solution is multiplied by 2. If the value type is integer, then the dual solution is scaled by 4.

Template Parameters
GRThe undirected graph type the algorithm runs on.
WMThe type edge weight map. The default type is GR::EdgeMap<int>.

#include <lemon/fractional_matching.h>

Public Types

typedef GR Graph
 The graph type of the algorithm.
 
typedef WM WeightMap
 The type of the edge weight map.
 
typedef WeightMap::Value Value
 The value type of the edge weights.
 
typedef Graph::template
NodeMap< typename Graph::Arc > 
MatchingMap
 The type of the matching map.
 

Public Member Functions

 MaxWeightedFractionalMatching (const Graph &graph, const WeightMap &weight, bool allow_loops=true)
 
Execution Control

The simplest way to execute the algorithm is to use the run() member function.

void init ()
 Initialize the algorithm.
 
void start ()
 Start the algorithm.
 
void run ()
 Run the algorithm.
 
Primal Solution

Functions to get the primal solution, i.e. the maximum weighted matching.
Either run() or start() function should be called before using them.

Value matchingWeight () const
 Return the weight of the matching.
 
int matchingSize () const
 Return the number of covered nodes in the matching.
 
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.
 
const MatchingMapmatchingMap () const
 Return a const reference to the matching map.
 
Dual Solution

Functions to get the dual solution.
Either run() or start() function should be called before using them.

Value dualValue () const
 Return the value of the dual solution.
 
Value nodeValue (const Node &n) const
 Return the dual value (potential) of the given node.
 

Static Public Attributes

static const int primalScale = 2
 
static const int dualScale
 Scaling factor for dual solution.
 

Constructor & Destructor Documentation

MaxWeightedFractionalMatching ( const Graph graph,
const WeightMap weight,
bool  allow_loops = true 
)
inline

Constructor.

Member Function Documentation

void init ( )
inline

This function initializes the algorithm.

void start ( )
inline

This function starts the algorithm.

Precondition
init() must be called before using this function.
void run ( )
inline

This method runs the MaxWeightedFractionalMatching algorithm.

Note
mwfm.run() is just a shortcut of the following code.
mwfm.init();
mwfm.start();
Value matchingWeight ( ) const
inline

This function returns the weight of the found matching. This value is scaled by primal scale.

Precondition
Either run() or start() must be called before using this function.
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.
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.
const MatchingMap& matchingMap ( ) const
inline

This function returns a const reference to a node map that stores the matching arc (or edge) incident to each node.

Value dualValue ( ) const
inline

This function returns the value of the dual solution. It should be equal to the primal value scaled by dual scale.

Precondition
Either run() or start() must be called before using this function.
Value nodeValue ( const Node &  n) const
inline

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

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

Member Data Documentation

const int primalScale = 2
static

Scaling factor for primal solution.

const int dualScale
static
Initial value:

Scaling factor for dual solution. It is equal to 4 or 1 according to the value type.