Sum of Squares Lower Bounds from Pairwise Independence

Abstract

We prove that for every ε>0 and predicate P:{0,1}k-> {0,1} that supports a pairwise independent distribution, there exists an instance I of the Max P constraint satisfaction problem on n variables such that no assignment can satisfy more than a ~(|P-1(1)|)/(2k)+ε fraction of I's constraints but the degree Ω(n) Sum of Squares semidefinite programming hierarchy cannot certify that I is unsatisfiable. Similar results were previously only known for weaker hierarchies.