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

Susanna F. de Rezende, Aaron Potechin, Kilian Risse

TL;DR

This work proves an unconditional average-case lower bound for Sherali-Adams with polynomially bounded coefficients on the unary k-clique encoding: for $D\le 2\log n$ and $k\le n^{1/66}$, refuting the $n^{1/100}$-clique formula on $G\sim\mathcal{G}(n,k,n^{-2/D})$ requires refutations of size $n^{Ω(D)}$, matching up to constants in the exponent. The authors introduce a pseudo-measure $\mu_d$ on monomials, inspired by pseudo-calibration, which quantifies progress of a refutation and, via duality, lower-bounds the required coefficient size. They organize the analysis around cores and boundary families, proving that random graphs are asymptotically well-behaved and that the measure concentrates on a small family of “good” rectangles where $\mu_d$ is essentially positive; edge-axiom subrectangles contribute negligibly. The technique yields a tight average-case bound for Sherali-Adams and extends to related semi-algebraic proofs, marking a first size lower bound for the clique formula in such systems and suggesting potential extensions to other proof frameworks and to more general random graph regimes.

Abstract

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size $n^{Ω(d)}$ to rule out the existence of an $n^{Θ(1)}$-clique in Erdős-Rényi 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.

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

TL;DR

This work proves an unconditional average-case lower bound for Sherali-Adams with polynomially bounded coefficients on the unary k-clique encoding: for and , refuting the -clique formula on requires refutations of size , matching up to constants in the exponent. The authors introduce a pseudo-measure on monomials, inspired by pseudo-calibration, which quantifies progress of a refutation and, via duality, lower-bounds the required coefficient size. They organize the analysis around cores and boundary families, proving that random graphs are asymptotically well-behaved and that the measure concentrates on a small family of “good” rectangles where is essentially positive; edge-axiom subrectangles contribute negligibly. The technique yields a tight average-case bound for Sherali-Adams and extends to related semi-algebraic proofs, marking a first size lower bound for the clique formula in such systems and suggesting potential extensions to other proof frameworks and to more general random graph regimes.

Abstract

We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size to rule out the existence of an -clique in Erdős-Rényi random graphs whose maximum clique is of size . 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.
Paper Structure (28 sections, 44 theorems, 107 equations, 3 figures, 1 algorithm)

This paper contains 28 sections, 44 theorems, 107 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Result
  3. Organization
  4. Preliminaries
  5. Semantic Sherali-Adams
  6. Graph Theory
  7. Clique Formula
  8. Main Theorem and Proof Overview
  9. Our Pseudo-Measure
  10. Expected Behavior of Our Pseudo-Measure
  11. Edge Axioms Should Have Small Measure
  12. Rectangles Should Be Approximately Non-Negative
  13. Cores
  14. Cores and Boundaries
  15. Proof of \ref{['lem:compression']}
  16. ...and 13 more sections

Key Result

Theorem 1

For all integers $n \in \mathbb{N}^+$ and $D \le 2 \log n$, if $G \sim \mathcal{G}(n,n^{-2/D})$ is an Erdős-Rényi random graph, then it holds asymptotically almost surely that Sherali-Adams with polynomially bounded coefficients requires size at least $n^{\Omega(D)}$ to refute the claim that $G$ con

Figures (3)

  • Figure 1: A pattern graph $H$ mapped onto a tuple $t=(v_1, \ldots, v_k)$
  • Figure 2: A core with edges in $M_1$ and $M_2$ highlighted
  • Figure 3: The rectangle $Q$ is a good rectangles as the vertices in $R$ have all vertices as neighbors, the blocks outside $R$ are large and small tuples on these blocks have common neighborhoods of expected size

Theorems & Definitions (97)

  • Theorem 1: Informal
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Lemma 5
  • proof
  • Proposition 6
  • ...and 87 more