Shortest Reconfiguration of Matchings

Imagine that unlabelled tokens are placed on edges forming a matching of a graph. A token can be moved to another edge provided that the edges containing tokens remain a matching. The distance between two configurations of tokens is the minimum number of moves required to transform one into the other. We study the problem of computing the distance between two given configurations. We prove that if source and target configurations are maximal matchings, then the problem admits no polynomial-time sublogarithmic-factor approximation algorithm unless \(\mathsf{P}= \mathsf{NP}\). On the positive side, we show that for matchings of bipartite graphs the problem is fixed-parameter tractable parameterized by the size d of the symmetric difference of the two given configurations. Furthermore, we obtain a \(d^\varepsilon \)-factor approximation algorithm for the distance of two maximum matchings of bipartite graphs for every \(\varepsilon > 0\). The proofs of our positive results are constructive and can hence be turned into algorithms that output shortest transformations. Both algorithmic results rely on a close connection to the Directed Steiner Tree problem. Finally, we show that determining the exact distance between two configurations is complete for the class \(\mathsf{D}^\mathsf{P}\), and determining the maximum distance between any two configurations of a given graph is \(\mathsf{D}^\mathsf{P}\)-hard.