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 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 o...More