Tight Sherali-Adams Lower Bounds for the Average-Case k-Clique Problem on Erdős-Rényi Random Graphs

Core Concepts

Sherali-Adams with polynomially bounded coefficients requires size nΩ(D) to refute the existence of an nΘ(1)-clique in Erdős-Rényi random graphs whose maximum clique size is at most D ≤ 2 log n.

Abstract

The paper establishes a tight, up to constants in the exponent, Sherali-Adams coefficient size lower bound for k-clique formulas over Erdős-Rényi random graphs. The key insights are: The authors introduce a new technique, inspired by pseudo-calibration, to define a pseudo-measure that precisely captures the contribution of a monomial to a Sherali-Adams refutation. This pseudo-measure intuitively captures progress and should have further applications in proof complexity. The authors show that for Erdős-Rényi random graphs G ~ G(n, k, 1/2) with maximum clique size D ≤ 2 log n and k ≤ n^(1/66), Sherali-Adams with polynomially bounded coefficients requires size nΩ(D) to refute the k-clique formula over G. The proof strategy involves defining the notion of a "core" of a graph, which captures the combinatorial structure needed for the lower bound, and showing that random graphs are "well-behaved" with respect to this notion. A key technical challenge is bounding the measure of edge axioms and showing that the pseudo-measure is concentrated on "good" rectangles. The authors overcome this using a careful analysis of the Fourier expansion of the pseudo-measure.

Stats

The maximum clique size in the random graph G ~ G(n, k, 1/2) is at most D ≤ 2 log n. The number of vertices in the graph is n, and the number of blocks in the k-partite graph is k ≤ n^(1/66).

Quotes

"We prove that Sherali-Adams with polynomially bounded coefficients requires proofs of size nΩ(d) to rule out the existence of an nΘ(1)-clique in Erdős-Rényi random graphs whose maximum clique is of size d ⩽2 log n." "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."

Key Insights Distilled From

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

by Susanna F. d... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16722.pdf

Deeper Inquiries

What other proof systems or models of computation could benefit from the pseudo-measure technique introduced in this work

The pseudo-measure technique introduced in this work could potentially benefit other proof systems or models of computation that involve analyzing the contribution of monomials to a refutation. For example, it could be applied to semi-algebraic proof systems like Sum-of-Squares or Nullstellensatz, where understanding the impact of individual monomials on the refutation size is crucial. Additionally, the technique could be useful in analyzing the complexity of algorithms based on certificates of unsatisfiability in different proof systems.

Can the lower bound be extended to larger values of the maximum clique size D, beyond the 2 log n regime

The lower bound presented in this work for the maximum clique size D could potentially be extended to larger values beyond the 2 log n regime. By refining the pseudo-measure technique and adjusting the analysis to accommodate larger values of D, it may be possible to establish tighter lower bounds for a wider range of maximum clique sizes. This extension could provide further insights into the average-case hardness of k-clique problems on random graphs with varying maximum clique sizes.

How might the insights from this work on average-case hardness of k-clique translate to other average-case complexity questions in computer science

The insights from this work on the average-case hardness of k-clique could be applied to other average-case complexity questions in computer science. For example, similar techniques could be used to study the average-case complexity of other combinatorial problems, such as graph coloring, Hamiltonian cycles, or SAT solving. By adapting the pseudo-measure technique and proof strategies to different problem domains, researchers could gain a deeper understanding of the average-case complexity landscape and potentially uncover new hardness results for various computational problems.