Refuting Perfect Matchings in Spectral Expanders is Hard

Ari Biswas; Rajko Nenadov

Refuting Perfect Matchings in Spectral Expanders is Hard

Combinatorics 2026-02-12 v2 Computational Complexity

Authors: Ari Biswas , Rajko Nenadov

Abstract

This work studies the complexity of refuting the existence of a perfect matching in spectral expanders with an odd number of vertices, in the Polynomial Calculus (PC) and Sum of Squares (SoS) proof system. Austrin and Risse [SODA, 2021] showed that refuting perfect matchings in sparse $d$ -regular \emph{random} graphs, in the above proof systems, with high probability requires proofs with degree $\Omega(n/\log n)$ . We extend their result by showing the same lower bound holds for \emph{all} $d$ -regular graphs with a mild spectral gap.

Keywords

spectral theory bound random graphs

Cite

@article{arxiv.2506.07700,
  title  = {Refuting Perfect Matchings in Spectral Expanders is Hard},
  author = {Ari Biswas and Rajko Nenadov},
  journal= {arXiv preprint arXiv:2506.07700},
  year   = {2026}
}

Comments

Improved presentation, fixed minor issue in Proofs of Lemma 4.1 and 4.2. Accepted at SIAM Journal For Discrete Mathematics

Refuting Perfect Matchings in Spectral Expanders is Hard

Abstract

Keywords

Cite

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