Simultaneous matchings: Hardness and approximation

Abstract

Given a bipartite graph G=(X∪˙ D, E⊆ X× D), an X-perfect matching is a matching in G that covers every node in X. In this paper we study the following generalisation of the X-perfect matching problem, which has applications in constraint programming: Given a bipartite graph as above and a collection F⊆ 2 X of k subsets of X, find a subset M⊆ E of the edges such that for each C∈ F, the edge set M∩(C× D) is a C-perfect matching in G (or report that no such set exists). We show that the decision problem is NP-complete and that the corresponding optimisation problem is in APX when k= O (1) and even APX-complete already for k= 2. On the positive side, we show that a 2/(k+ 1)-approximation can be found in poly (k,| X∪ D|) time. We show also that such an approximation M can be found in time (k+(k 2) 2 k− 2) poly (| X∪ D|), with the further restriction that each vertex in D has degree at most 2 in M.