Simultaneous matchings: Hardness and approximation

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 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)poly(|X ∪D|), with the further restriction that each vertex in D has degree at most 2 in M .