On the Complexity of Optimal Matching Reconfiguration
We consider the problem of matching reconfiguration, where we are given two matchings \(M_s\) and \(M_t\) in a graph G and the goal is to determine if there exists a sequence of matchings \(M_0, M_1, \ldots , M_\ell \), such that \(M_0 = M_s\), all consecutive matchings differ by exactly two edges (specifically, any matching is obtained from the previous one by the addition and deletion of one edge), and \(M_\ell = M_t\). It is known that the existence of such a sequence can be determined in polynomial time .
We extend the study of reconfiguring matchings to account for the length of the reconfiguration sequence. We show that checking if we can reconfigure \(M_s\) to \(M_t\) in at most \(\ell \) steps is NP-hard, even when the graph is unweighted, bipartite, and the maximum degree is four, and the matchings \(M_s\) and \(M_t\) are maximum matchings. We propose two simple algorithmic approaches, one of which improves on the brute-force running time while the other is a SAT formulation that we expect will be useful in practice.
KeywordsGraph theory Reconfiguration Matchings NP-hardness
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