Solving Matching Problems Efficiently in Bipartite Graphs

Abstract

We investigate the problem maxDMM of computing a largest set of pairwise disjoint maximum matchings in undirected graphs. In this paper, n, m denote, respectively, the number of vertices and the number of edges. We solve maxDMM for bipartite graphs, by providing an $O(n^{1.5}\sqrt{m/\log n} + mn\log n)$-time algorithm. We design better algorithms for complete bipartite graphs, and bisplit graphs. (Bisplit graphs are bipartite graphs with the nested neighborhood property.) Specifically, we prove that the problem maxDMM is solvable in complete bipartite graphs in time O(m). A sequence $S=(s_1,\cdots ,s_t)$ of positive integers is said to be color-feasible for a graph G, if there exists a proper edge-coloring of G with colors $1,\cdots ,t$, such that precisely $s_i$ edges have color i, for every $i=1,\cdots ,t$. Actually, for complete bipartite graphs, we prove that, for any sequence S of integers which is color-feasible for a complete bipartite graph G, an edge-coloring of G corresponding to S can be obtained in time O(m). For bisplit graphs, (1) we solve maxDMM in time $O(mn\log n)$, and (2) we design an $O(n^2\log n)$-time algorithm to count all maximum matchings. This latter time is the same time in which runs the best known algorithm computing the number of maximum matchings in bisplit graphs [17], but our algorithm is much simpler than the one given in [17]. The key idea underlying both results is that bisplit graphs have an O(n)-time enumeration of their minimal vertex covers.