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arXiv · 2506.07700

Refuting Perfect Matchings in Spectral Expanders is Hard

Ari Biswas·Rajko Nenadov
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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 $Ω(n/\log n)$. We extend their result by showing the same lower bound holds for \emph{all} $d$-regular graphs with a mild spectral gap.

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Ari Biswas, Rajko Nenadov. 2026-02-11. Refuting Perfect Matchings in Spectral Expanders is Hard. https://arxiv.org/abs/2506.07700

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