Clique Is Hard on Average for Sherali-Adams with Bounded Coefficients

TL;DR

This paper establishes tight lower bounds on the size of Sherali-Adams proofs with bounded coefficients for detecting large cliques in Erdős-Rényi graphs, introducing a novel pseudo-calibration inspired technique.

Contribution

It proves tight lower bounds for Sherali-Adams proofs on clique problems and introduces a new pseudo-calibration based method for analyzing proof complexity.

Findings

Sherali-Adams with polynomially bounded coefficients requires proofs of size n^{Ω(d)}.

The lower bound matches the upper bound up to a constant factor in the exponent.

The introduced pseudo-calibration technique may have broader applications in proof complexity.

Abstract

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{\Omega(d)}$ to rule out the existence of an $n^{\Theta(1)}$-clique in Erd\H{o}s-R\'{e}nyi random graphs whose maximum clique is of size $d\leq 2\log n$. This lower bound is tight up to the multiplicative constant in the exponent. We obtain this result by introducing a technique inspired by pseudo-calibration which may be of independent interest. The technique involves defining a measure on monomials that precisely captures the contribution of a monomial to a refutation. This measure intuitively captures progress and should have further applications in proof complexity.