A Generalized Matching Reconfiguration Problem

TL;DR

This paper introduces a generalized matching reconfiguration problem allowing multiple edge changes per step, providing optimal algorithms for unweighted and weighted cases, with applications to dynamic graph matchings and improved recourse bounds.

Contribution

It generalizes the classical matching reconfiguration problem by allowing multiple changes per step and offers optimal algorithms for both unweighted and weighted matchings.

Findings

Optimal transformation procedure for unweighted matchings with Δ=3

Asymptotically optimal algorithms for weighted matchings

Application to dynamic graph matchings with improved recourse bounds

Abstract

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 [...]