Higher Hardness Results for the Reconfiguration of Odd Matchings

TL;DR

This paper investigates the computational complexity of reconfiguring odd matchings in graphs, proving hardness results for diameter, radius, and shortest flip sequence problems, and establishing approximation limits.

Contribution

It establishes new complexity bounds for the reconfiguration of odd matchings, including hardness of computing diameter, radius, and shortest flip sequences, and shows approximation hardness.

Findings

Computing the diameter of the flip graph of odd matchings is $ ext{Pi}_2^p$-hard.

Computing the radius of the flip graph of odd matchings is $ ext{Sigma}_3^p$-hard.

Finding shortest flip sequences is $ ext{log}$-APX-hard and cannot be approximated within a sublogarithmic factor.

Abstract

We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and, consequently, isolates another vertex. The flip graph of odd matchings is a graph that has all odd matchings of a graph as vertices and an edge between two vertices if their corresponding matchings can be transformed into one another via a single flip. We show that computing the diameter of the flip graph of odd matchings is $\Pi_2^p$-hard. This complements a recent result by Wulf [FOCS25] that it is~$\Pi_2^p$-hard to compute the diameter of the flip graph of perfect matchings where a flip swaps matching edges along a single cycle of unbounded size. Further, we show that computing the radius of the flip graph of odd matchings is $\Sigma_3^p$-hard. The respective decision problems for the diameter and the radius are also complete in the respective level of the polynomial hierarchy. This shows that computing the radius of the flip graph of odd matchings is provably harder than computing its diameter, unless the polynomial hierarchy collapses. Finally, we reduce set cover to the problem of finding shortest flip sequences. As a consequence, we show $\log$-\APX-hardness and that the problem cannot be approximated by a sublogarithmic factor. By doing so, we answer a question asked by Aichholzer, Brenner, Dorfer, Hoang, Perz, Rieck, and Verciani [GD25].