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Higher Hardness Results for the Reconfiguration of Odd Matchings

Joseph Dorfer

TL;DR

This work analyzes the reconfiguration of combinatorial odd matchings through flips, formalizing a flip graph whose nodes are odd matchings and edges correspond to single flips. It delivers three main hardness results: computing the flip graph diameter is $\Pi_2^p$-complete, computing the radius is $\Sigma_3^p$-complete (and strictly harder than diameter unless the polynomial hierarchy collapses), and computing flip distances is $\log$-hard to approximate via a Set Cover reduction. The authors employ gadget-based reductions from quantified Boolean formulas and Set Cover, anchored by a detailed union-of-odd-matchings framework to bound flip sequences. These results significantly advance our understanding of how central structural properties and distance metrics in flip graphs behave at higher levels of the polynomial hierarchy, and they answer questions about approximation hardness in this reconfiguration setting.

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 $Π_2^p$-hard. This complements a recent result by Wulf [FOCS25] that it is~$Π_2^p$-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 $Σ_3^p$-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].

Higher Hardness Results for the Reconfiguration of Odd Matchings

TL;DR

This work analyzes the reconfiguration of combinatorial odd matchings through flips, formalizing a flip graph whose nodes are odd matchings and edges correspond to single flips. It delivers three main hardness results: computing the flip graph diameter is -complete, computing the radius is -complete (and strictly harder than diameter unless the polynomial hierarchy collapses), and computing flip distances is -hard to approximate via a Set Cover reduction. The authors employ gadget-based reductions from quantified Boolean formulas and Set Cover, anchored by a detailed union-of-odd-matchings framework to bound flip sequences. These results significantly advance our understanding of how central structural properties and distance metrics in flip graphs behave at higher levels of the polynomial hierarchy, and they answer questions about approximation hardness in this reconfiguration setting.

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 -hard. This complements a recent result by Wulf [FOCS25] that it is~-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 -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 -\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].
Paper Structure (10 sections, 3 theorems, 2 equations, 3 figures)

This paper contains 10 sections, 3 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Computing Central Structures in Flip Graphs
  4. Our Contributions
  5. Preliminaries
  6. Union of Odd Matchings
  7. The Polynomial Hierarchy
  8. APX-hardness
  9. Set Cover
  10. Computing the Diameter is $\Pi_2^p$-complete

Key Result

Theorem 1

Given a graph $G$ and a parameter $k\in \mathbb{N}$. Deciding whether the diameter of the flip graph of odd matchings of $G$ is at most $k$ is $\Pi_2^p$-complete.

Figures (3)

  • Figure 1: Illustration of a flip graph of odd matchings with a flip sequence of length $3$ highlighted.
  • Figure 2: Switching a cycle with three matching edges in four flips.
  • Figure 3: Gadgets of the reduction

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3