International Workshop on Combinatorial Algorithms

IWOCA 2014: Combinatorial Algorithms pp 128-139 | Cite as

Solving Matching Problems Efficiently in Bipartite Graphs

  • Selma DjelloulEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8986)


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.



I thank Odile Favaron for her helpful idea to handle the proof of Theorem 3 by arithmetic. I thank Pierre Fraigniaud for his careful reading and advices to improve the paper writing.


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© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.LRI, UMR 8623Orsay CedexFrance

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