Refuting Perfect Matchings in Spectral Expanders is Hard
Ari Biswas, Rajko Nenadov
TL;DR
The paper investigates the hardness of refuting the existence of a perfect matching in $d$-regular spectral expander graphs with an odd number of vertices within Polynomial Calculus and Sum-of-Squares. It extends prior results for random graphs by proving degree lower bounds of $\Omega\Big(\frac{n}{\log n}\Big)$ for refuting $\text{PM}(G)$ in all $(n,d,\lambda)$-graphs with a small spectral gap, via affine restrictions and a topological embedding of a hard instance $H$ into $G$. A key technical ingredient is a topological embedding theorem guaranteeing odd-length subdivisions of $H$ inside $G$, paired with a Tutte-based argument to ensure the residual graph supports the required regular subgraph or matching. This construction transfers the hardness from $H$ to $G$, yielding exponential size lower bounds in $n$ due to known degree-to-size relations. The work thus broadens the understanding of algebraic proof complexity for matching-type constraints on expanders and points to open questions about the tightness of the bounds and potential improvements.
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.