Refuting Perfect Matchings in Spectral Expanders is Hard

TL;DR

This paper proves that refuting perfect matchings in spectral expanders with an odd number of vertices is computationally hard within certain proof systems, extending previous results from random graphs to all such expanders.

Contribution

It establishes degree lower bounds for refutation in Polynomial Calculus and Sum of Squares proof systems for all $d$-regular spectral expanders, not just random graphs.

Findings

Refutation requires high-degree proofs in PC and SoS systems.

Lower bounds apply to all $d$-regular spectral expanders with spectral gap.

Extends previous results from random graphs to deterministic spectral expanders.

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.