Discussiones Mathematicae Graph Theory 33(3) (2013) 559-569
doi: 10.7151/dmgt.1694

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Maximum semi-matching problem in bipartite graphs

Ján Katrenič and Gabriel Semanišin

Institute of Computer Science
P.J. Šafárik University, Faculty of Science
Jesenná 5, 041 54 Košice, Slovak Republic

Abstract

An (f,g)-semi-matching in a bipartite graph G = (U ∪V,E) is a set of edges M ⊆ E such that each vertex u ∈ U is incident with at most f(u) edges of M, and each vertex v ∈ V is incident with at most g(v) edges of M. In this paper we give an algorithm that for a graph with n vertices and m edges, n ≤ m, constructs a maximum (f,g)-semi-matching in running time O(m·min{ √{ ∑u ∈ Uf(u)}, √{ ∑v ∈ Vg(v)}} ). Using the reduction of [5] our result on maximum (f,g)-semi-matching problem directly implies an algorithm for the optimal semi-matching problem with running time O( √nm logn).

Keywords: semi-matching, quasi-matching, bipartite graph, computational complexity

2010 Mathematics Subject Classification: 05C85, 05C70, 68Q17.

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Received 2 February 2012
Revised 25 June 2012
Accepted 25 June 2012