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

© 2019 Society for Industrial and Applied Mathematics We prove that with high probability over the choice of a random graph G from the Erd\H os-Rényi distribution G(n, 1/2), the nO(d)-time degree d sum-of-squares (SOS) semidefinite programming relaxation for the clique problem will give a value of at least n1/2 - c(d/ log n)1/2 for some constant c > 0. This yields a nearly tight n1/2 - 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