Sum of Squares Lower Bounds from Pairwise Independence

Abstract

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

27 Pages (including the title page) and 4 figures including appendix