Maximum semi-matching problem in bipartite graphs

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∈U f(u), √∑ v∈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 O( √ nm logn).