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Higher Hardness Results for the Reconfiguration of Odd Matchings

Author Joseph Dorfer

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LIPIcs.STACS.2026.33.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Discrete mathematics
Keywords
  • Graph Reconfiguration Problems
  • Flip Graphs
  • Polynomial Hierarchy
  • APX-hardness

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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 Π₂^p-hard. This complements a recent result by Wulf [FOCS25] that it is Π₂^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 Σ₃^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 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].

Cite As Get BibTex

Joseph Dorfer. Higher Hardness Results for the Reconfiguration of Odd Matchings. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.33

Author Details

Joseph Dorfer
  • Graz University of Technology, Austria

Funding

Austrian Science Foundation (FWF) 10.55776/DOC183.

Acknowledgements

First, I would like to thank Lasse Wulf for introducing me to the relation between the polynomial hierarchy and diameter computation for flip graphs. Further, I would like to thank Oswin Aichholzer, Sofia Brenner, Hung Hoang, Daniel Perz, Christian Rieck, and Francesco Verciani for our previous joint work on odd matchings. Finally, I want to thank the reviewers for many helpful comments that helped improving the argumentation throughout the paper.

References

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