Simultaneous matchings: Hardness and approximation

Given a bipartite graph G=([email protected][email protected]?D,[email protected]?XxD), 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 [email protected]?2^X of k subsets of X, find a subset [email protected]?E of the edges such that for each [email protected]?F, the edge set [email protected]?(CxD) 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,|[email protected]?D|) time. We show also that such an approximation M can be found in time (k+(k2)2^k^-^2)poly(|[email protected]?D|), with the further restriction that each vertex in D has degree at most 2 in M.