SDPs and Robust Satisfiability of Promise CSP

TL;DR

This paper explores the extension of robust satisfaction algorithms from CSPs to promise CSPs (PCSPs) using semidefinite programming, identifying conditions under which these algorithms are effective and establishing hardness results.

Contribution

It introduces robust SDP rounding algorithms for PCSPs based on high-dimensional symmetries and characterizes when such problems are computationally hard, advancing the theoretical understanding of PCSPs.

Findings

Robust SDP rounding algorithms work under certain symmetry conditions.

Lack of symmetries makes PCSPs hard for all symmetric Boolean predicates.

Connections established between SDP integrality gaps and sphere Ramsey theory.

Abstract

For a constraint satisfaction problem (CSP), a robust satisfaction algorithm is one that outputs an assignment satisfying most of the constraints on instances that are near-satisfiable. It is known that the CSPs that admit efficient robust satisfaction algorithms are precisely those of bounded width, i.e., CSPs whose satisfiability can be checked by a simple local consistency algorithm (eg., 2-SAT or Horn-SAT in the Boolean case). While the exact satisfiability of a bounded width CSP can be checked by combinatorial algorithms, the robust algorithm is based on rounding a canonical Semidefinite Programming (SDP) relaxation. In this work, we initiate the study of robust satisfaction algorithms for promise CSPs, which are a vast generalization of CSPs that have received much attention recently. The motivation is to extend the theory beyond CSPs, as well as to better understand the power of SDPs. We present robust SDP rounding algorithms under some general conditions, namely the existence of particular high-dimensional Boolean symmetries known as majority or alternating threshold polymorphisms. On the hardness front, we prove that the lack of such polymorphisms makes the PCSP hard for all pairs of symmetric Boolean predicates. Our approach relies on SDP integrality gaps argued via the absence of certain colorings of the sphere, with connections to sphere Ramsey theory. We conjecture that PCSPs with robust satisfaction algorithms are precisely those for which the feasibility of the canonical SDP implies (exact) satisfiability. We also give a precise algebraic condition, known as a minion characterization, of which PCSPs have the latter property.