Sum of Squares Lower Bounds from Pairwise Independence

TL;DR

This paper demonstrates that for certain predicates supporting pairwise independent distributions, the Sum of Squares hierarchy cannot certify unsatisfiability even when the instance is nearly unsatisfiable, extending previous weaker hierarchy results.

Contribution

It proves that the Sum of Squares hierarchy has limitations in certifying unsatisfiability for MaxP problems with pairwise independent predicates, for all epsilon > 0.

Findings

Sum of Squares hierarchy cannot certify unsatisfiability for certain MaxP instances.

Results extend previous limitations known for weaker hierarchies.

Instances are nearly unsatisfiable but remain hard for high-degree SOS proofs.

Abstract

We prove that for every $\epsilon>0$ and predicate $P:\{0,1\}^k\rightarrow \{0,1\}$ 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 $\tfrac{|P^{-1}(1)|}{2^k}+\epsilon$ fraction of $\mathcal{I}$'s constraints but the degree $\Omega(n)$ Sum of Squares semidefinite programming hierarchy cannot certify that $\mathcal{I}$ is unsatisfiable. Similar results were previously only known for weaker hierarchies.