Abstract
Given a bipartite graph \(G = (X \dot{\cup} D,E \subseteq X \times D)\), 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 \(\mathcal{F} \subseteq 2^{X}\) of k subsets of X, find a subset M ⊆ E of the edges such that for each \(C \in \mathcal{F}\), 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 2kpoly(k,|X ∪ D|) time.
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Elbassioni, K., Katriel, I., Kutz, M., Mahajan, M. (2005). Simultaneous Matchings. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_12
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DOI: https://doi.org/10.1007/11602613_12
Publisher Name: Springer, Berlin, Heidelberg
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