On Approximability of Satisfiable k-CSPs: V

TL;DR

This paper introduces a new framework for analyzing the approximability of satisfiable k-CSPs, combining hybrid algorithms and a novel invariance principle to extend prior theoretical results.

Contribution

It develops a hybrid approximation algorithm and a matching dictator vs. quasirandom test, along with the mixed invariance principle, advancing the understanding of Max-CSPs.

Findings

Proposes a new hybrid approximation algorithm for Max-CSPs.

Introduces the mixed invariance principle extending Mossel et al.'s work.

Provides a matching dictator vs. quasirandom test with perfect completeness.

Abstract

We propose a framework of algorithm vs. hardness for all Max-CSPs and demonstrate it for a large class of predicates. This framework extends the work of Raghavendra [STOC, 2008], who showed a similar result for almost satisfiable Max-CSPs. Our framework is based on a new hybrid approximation algorithm, which uses a combination of the Gaussian elimination technique (i.e., solving a system of linear equations over an Abelian group) and the semidefinite programming relaxation. We complement our algorithm with a matching dictator vs. quasirandom test that has perfect completeness. The analysis of our dictator vs. quasirandom test is based on a novel invariance principle, which we call the mixed invariance principle. Our mixed invariance principle is an extension of the invariance principle of Mossel, O'Donnell and Oleszkiewicz [Annals of Mathematics, 2010] which plays a crucial role in Raghavendra's work. The mixed invariance principle allows one to relate 3-wise correlations over discrete probability spaces with expectations over spaces that are a mixture of Guassian spaces and Abelian groups, and may be of independent interest.