A NEARLY TIGHT SUM-OF-SQUARES LOWER BOUND FOR THE PLANTED CLIQUE PROBLEM

We prove that with high probability over the choice of a random graph G from the Erdos-Renyi 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 )or 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(logn). 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.