Approximation Resistant Predicates From Pairwise Independence

TL;DR

This paper establishes a new sufficient condition for approximation resistance of predicates on multiple variables over a finite domain, based on the existence of a specific pairwise independent distribution within the satisfying assignments, under the Unique Games Conjecture.

Contribution

It introduces a novel criterion linking pairwise independence in distributions to approximation resistance of predicates, advancing understanding in computational complexity.

Findings

Predicate $P$ is approximation resistant if a balanced pairwise independent distribution exists within its satisfying assignments.

Provides a new sufficient condition for approximation resistance under the Unique Games Conjecture.

Enhances theoretical framework for analyzing the hardness of approximation problems.

Abstract

We study the approximability of predicates on $k$ variables from a domain $[q]$, and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate $P$ is approximation resistant if there exists a balanced pairwise independent distribution over $[q]^k$ whose support is contained in the set of satisfying assignments to $P$.