A Nearly Tight Sum-of-Squares Lower Bound for the Planted Clique Problem

TL;DR

This paper establishes a nearly tight lower bound for the Sum-of-Squares relaxation in the planted clique problem, demonstrating limitations of this approach for certain graph sizes, and introduces a new pseudo-calibration framework for constructing such bounds.

Contribution

It provides a nearly tight lower bound for the Sum-of-Squares relaxation in the planted clique problem and introduces the pseudo-calibration framework for Sum-of-Squares lower bounds.

Findings

Sum-of-Squares relaxation yields high values for certain graph sizes

Nearly tight $n^{1/2 - o(1)}$ bound established

Introduces pseudo-calibration framework for lower bounds

Abstract

We prove that with high probability over the choice of a random graph $G$ from the Erd\H{o}s-R\'enyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least $n^{1/2-c(d/\log n)^{1/2}}$ for some constant $c>0$. This yields a nearly tight $n^{1/2 - o(1)}$ bound on the value of this program for any degree $d = o(\log n)$. Moreover we introduce a new framework that we call \emph{pseudo-calibration} to construct Sum of Squares lower bounds. This framework is inspired by taking a computational analog of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.