Elsevier

Theoretical Computer Science

Volume 423, 16 March 2012, Pages 50-58
Theoretical Computer Science

Finding all maximally-matchable edges in a bipartite graph

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Abstract

We consider the problem of finding all maximally-matchable edges in a bipartite graph G=(V,E), i.e., all edges that are included in some maximum matching. We show that given any maximum matching in the graph, it is possible to perform this computation in linear time O(n+m) (where n=|V| and m=|E|). Hence, the time complexity of finding all maximally-matchable edges reduces to that of finding a single maximum matching, which is O(n1/2m) (Hopcroft and Karp [12]), or O((n/logn)1/2m) for dense graphs with m=Θ(n2) (Alt et al. [2]). This time complexity improves upon that of the best known algorithms for the problem, which is O(nm) (Costa [5] for bipartite graphs, and Carvalho and Cheriyan [6] for general graphs). Other algorithms for solving that problem are randomized algorithms due to Rabin and Vazirani [15] and Cheriyan [3], the runtime of which is Õ(n2.376). Our algorithm, apart from being deterministic, improves upon that time complexity for bipartite graphs when m=O(nr) and r<1.876. In addition, our algorithm is elementary, conceptually simple, and easy to implement.

Keywords

Bipartite graphs
Perfect matchings
Maximum matchings
Maximally-matchable edges

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