핵심 개념
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.
초록
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.
통계
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).
인용구
"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."