CLAP: A New Algorithm for Promise CSPs

TL;DR

This paper introduces CLAP, a novel algorithm for Promise CSPs that combines two relaxations, providing a new tractability criterion based on polymorphism symmetry, and solves problems beyond existing algorithms.

Contribution

The paper presents CLAP, a new algorithm for Promise CSPs, with a characterization of its power and a tractability condition based on polymorphism symmetry.

Findings

CLAP can solve PCSPs not solvable by previous algorithms.

A characterization of CLAP's power via minion homomorphism.

Identification of a weak symmetry condition guaranteeing tractability.

Abstract

We propose a new algorithm for Promise Constraint Satisfaction Problems PCSPs). It is a combination of the $\textbf{C}$onstraint Basic $\textbf{L}$P relaxation and the $\textbf{A}$ffine I$\textbf{P}$ relaxation (CLAP). We give a characterisation of the power of CLAP in terms of a minion homomorphism. Using this characterisation, we identify a certain weak notion of symmetry which, if satisfied by infinitely many polymorphisms of PCSPs, guarantees tractability. We demonstrate that there are PCSPs solved by CLAP that are not solved by any of the existing algorithms for PCSPs; in particular, not by the BLP+AIP algorithm of Brakensiek et al. [SICOMP'20] and not by a reduction to tractable finite-domain CSPs.