On the Complexity of Optimal Matching Reconfiguration

Abstract

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 [5].

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.