SOS lower bound for exact planted clique

Abstract

We prove a SOS degree lower bound for the planted clique problem on the Erdös-Rényi random graph G(n, 1/2). The bound we get is degree d = Ω(ϵ2 log n/ log log n) for clique size ω = n1/2−ϵ, which is almost tight. This improves the result of [5] for the "soft" version of the problem, where the family of the equality-axioms generated by x1 + ... + xn = ω is relaxed to one inequality x1 + ... + xn ≥ ω. As a technical by-product, we also "naturalize" certain techniques that were developed and used for the relaxed problem. This includes a new way to define the pseudo-expectation, and a more robust method to solve out the coarse diagonalization of the moment matrix.