This class provides an efficient implementation of Edmond's maximum weighted matching algorithm. The implementation is based on extensive use of priority queues and provides time complexity.
The maximum weighted matching problem is to find a subset of the edges in an undirected graph with maximum overall weight for which each node has at most one incident edge. It can be formulated with the following linear program.
where is the set of edges incident to a node in , is the set of edges with both ends in and is the set of odd cardinality subsets of the nodes.
The algorithm calculates an optimal 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.
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 and the BlossomIt nested class, which is able to iterate on the nodes of a blossom. If the value type is integer, then the dual solution is multiplied by 4.
GR | The undirected graph type the algorithm runs on. |
WM | The type edge weight map. The default type is GR::EdgeMap<int>. |
#include <lemon/matching.h>
Classes | |
class | BlossomIt |
Iterator for obtaining the nodes of a blossom. More... | |
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 | |
MaxWeightedMatching (const Graph &graph, const WeightMap &weight) | |
Constructor. More... | |
Execution Control | |
The simplest way to execute the algorithm is to use the run() member function. | |
void | init () |
Initialize the algorithm. More... | |
void | fractionalInit () |
Initialize the algorithm with fractional matching. More... | |
void | start () |
Start the algorithm. More... | |
void | run () |
Run the algorithm. More... | |
Primal Solution | |
Value | matchingWeight () const |
Return the weight of the matching. More... | |
int | matchingSize () const |
Return the size (cardinality) of the matching. More... | |
bool | matching (const Edge &edge) const |
Return true if the given edge is in the matching. More... | |
Arc | matching (const Node &node) const |
Return the matching arc (or edge) incident to the given node. More... | |
const MatchingMap & | matchingMap () const |
Return a const reference to the matching map. More... | |
Node | mate (const Node &node) const |
Return the mate of the given node. More... | |
Dual Solution | |
Value | dualValue () const |
Return the value of the dual solution. More... | |
Value | nodeValue (const Node &n) const |
Return the dual value (potential) of the given node. More... | |
int | blossomNum () const |
Return the number of the blossoms in the basis. More... | |
int | blossomSize (int k) const |
Return the number of the nodes in the given blossom. More... | |
Value | blossomValue (int k) const |
Return the dual value (ptential) of the given blossom. More... | |
Static Public Attributes | |
static const int | dualScale |
Scaling factor for dual solution. More... | |
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Constructor.
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This function initializes the algorithm.
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This function initializes the algorithm with a fractional matching. This initialization is also called jumpstart heuristic.
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This function starts the algorithm.
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This method runs the MaxWeightedMatching
algorithm.
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This function returns a const reference to a node map that stores the matching arc (or edge) incident to each node.
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This function returns the value of the dual solution. It should be equal to the primal value scaled by dual scale.
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Scaling factor for dual solution. It is equal to 4 or 1 according to the value type.