Perfect Matching in Random Graphs is as Hard as Tseitin

TL;DR

This paper demonstrates that proving the non-existence of perfect matchings in certain random graphs is computationally as hard as well-known complex proof systems, resolving a question by Razborov.

Contribution

It establishes lower bounds on proof complexity for specific graph properties in multiple proof systems, linking random graph problems to proof complexity theory.

Findings

Polynomial Calculus requires degree Ω(n / log n)

Sum-of-Squares proof system needs degree Ω(n / log n)

Bounded-depth Frege proofs are exponentially large

Abstract

We study the complexity of proving that a sparse random regular graph on an odd number of vertices does not have a perfect matching, and related problems involving each vertex being matched some pre-specified number of times. We show that this requires proofs of degree $\Omega(n / \log n)$ in the Polynomial Calculus (over fields of characteristic $\ne 2$) and Sum-of-Squares proof systems, and exponential size in the bounded-depth Frege proof system. This resolves a question by Razborov asking whether the Lov\'asz-Schrijver proof system requires $n^\delta$ rounds to refute these formulas for some $\delta > 0$. The results are obtained by a worst-case to average-case reduction of these formulas relying on a topological embedding theorem which may be of independent interest.