A Generalized Matching Reconfiguration Problem

The goal in {\em reconfiguration problems} is to compute a {\em gradual transformation} between two feasible solutions of a problem such that all intermediate solutions are also feasible. In the {\em Matching Reconfiguration Problem} (MRP), proposed in a pioneering work by Ito et al.\ from 2008, we are given a graph $G$ and two matchings $M$ and $M'$, and we are asked whether there is a sequence of matchings in $G$ starting with $M$ and ending at $M'$, each resulting from the previous one by either adding or deleting a single edge in $G$, without ever going through a matching of size $< \min\{|M|,|M'|\}-1$. Ito et al.\ gave a polynomial time algorithm for the problem. In this paper we introduce a natural generalization of the MRP that depends on an integer parameter $\Delta \ge 1$: here we are allowed to make $\Delta$ changes to the current solution rather than 1 at each step of the {transformation procedure}. There is always a valid sequence of matchings transforming $M$ to $M'$ if $\Delta$ is sufficiently large, and naturally we would like to minimize $\Delta$. We first devise an optimal transformation procedure for unweighted matching with $\Delta = 3$, and then extend it to weighted matchings to achieve asymptotically optimal guarantees. The running time of these procedures is linear. We further demonstrate the applicability of this generalized problem to dynamic graph matchings. In this area, the number of changes to the maintained matching per update step (the \emph{recourse bound}) is an important quality measure. Nevertheless, the \emph{worst-case} recourse bounds of almost all known dynamic matching algorithms are prohibitively large, much larger than the corresponding update times. We fill in this gap via a surprisingly simple black-box reduction: Any dynamic algorithm for maintaining [...]