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

Boaz Barak, Samuel B. Hopkins, Jonathan A. Kelner, Pravesh K. Kothari, Ankur Moitra, Aaron Potechin

Abstract: We prove that with high probability over the choice of a random graph $G$ from the Erdös--Rényi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ sum-of-squares (SOS) 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 pseudocalibration to construct SOS lower bounds. This framework is inspired by taking a computational analogue of Bayesian probability theory. It yields a general recipe for constructing good pseudodistributions (i.e., dual certificates for the SOS semidefinite program) and sheds further light on the ways in which this hierarchy differs from others.