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Maximum Cardinality Matchings in General Graphs ( mc_matching )

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| < = $ \lfloor$ni/2$ \rfloor$ and | N1| < = n1 and hence

$ \Vert N\Vert <= n_1 + \sum_{i >= 2} \lfloor n_i/2 \rfloor $
for any matching N and any odd-set cover OSC.

It can be shown that for a maximum cardinality matching M there is always an odd-set cover OSC with

$ \Vert M\Vert = n_1 + \sum_{i >= 2} \lfloor n_i/2 \rfloor, $
thus proving the optimality of M. In such a cover all ni with i > = 2 are odd, hence the name.

list<edge> MAX_CARD_MATCHING(const graph& G, node_array<int>& OSC, int heur = 0)
    computes a maximum cardinality matching M in G and returns it as a list of edges. The algorithm ([26], [38]) has running time O(nm*$ \alpha$(n, m)). With 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.


next up previous contents index
Next: General Weighted Matchings ( Up: Graph Algorithms Previous: Bipartite Weighted Matchings and   Contents   Index
root 2008-01-09