A matching in a graph G is a subset M of the edges of G such that no two share an endpoint.
An odd-set cover OSC of G is a labeling of the nodes of G with non-negative integers such that every edge of G (which is not a self-loop) is either incident to a node labeled 1 or connects two nodes labeled with the same i , i > = 2 .
Let ni be the number of nodes labeled i and consider any matching N . For i , i > = 2 , let Ni be the edges in N that connect two nodes labeled i . Let N1 be the remaining edges in N . Then | Ni| < = ni/2 and | N1| < = n1 and hence
It can be shown that for a maximum cardinality matching M there is always an odd-set cover OSC with
|list< edge>||MAX_CARD_MATCHING(const graph& G, node_array< int> & OSC, int heur = 0)|
|computes a maximum cardinality matching M
returns it as a list of edges.
The algorithm (, ) has running
heur = 1
the algorithm uses a greedy heuristic
to find an initial matching.
This seems to have little effect on the running time of the algorithm.
An odd-set cover that proves the maximality of M is returned in OSC.
|list< edge>||MAX_CARD_MATCHING(const graph& G, int heur = 0)|
|as above, but no proof of optimality is returned.|
|bool||CHECK_MAX_CARD_MATCHING(const graph& G, const list< edge> & M, const node_array< int> & OSC)|
|checks whether M is a maximum cardinality matching in G and OSC is a proof of optimality. Aborts if this is not the case.|