Maximum semi-matching problem in bipartite graphs

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 Ο(m · min{ Σu∈U f(u), Σv∈V g(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 Ο(√nmlog n).