Sum of Squares Lower Bounds from Pairwise Independence

Boaz Barak; Siu On Chan; Pravesh Kothari

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arXiv · 1501.00734

Sum of Squares Lower Bounds from Pairwise Independence

Boaz Barak·Siu On Chan·Pravesh Kothari

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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.

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Boaz Barak, Siu On Chan, Pravesh Kothari. 2015-01-04. Sum of Squares Lower Bounds from Pairwise Independence. https://arxiv.org/abs/1501.00734

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