Sandwiches for Promise Constraint Satisfaction

TL;DR

This paper explores the complexity of Promise Constraint Satisfaction Problems (PCSPs), providing new examples of PCSPs reducible to tractable CSPs of small size and analyzing their properties related to linear equations and polymorphisms.

Contribution

It presents the first example of a Boolean PCSP reducible to a small tractable CSP of size 3, and investigates properties related to linear systems and polymorphisms.

Findings

A Boolean PCSP reduces to a CSP over a structure of size 3 but not smaller.

Identification of PCSPs reducible to systems of linear equations.

Analysis of PCSPs with semilattice or majority polymorphisms.

Abstract

Promise Constraint Satisfaction Problems (PCSP) were proposed recently by Brakensiek and Guruswami arXiv:1704.01937 as a framework to study approximations for Constraint Satisfaction Problems (CSP). Informally a PCSP asks to distinguish between whether a given instance of a CSP has a solution or not even a specified relaxation can be satisfied. All currently known tractable PCSPs can be reduced in a natural way to tractable CSPs. Barto arXiv:1909.04878 presented an example of a PCSP over Boolean structures for which this reduction requires solving a CSP over an infinite structure. We give a first example of a PCSP over Boolean structures which reduces to a tractable CSP over a structure of size $3$ but not smaller. Further we investigate properties of PCSPs that reduce to systems of linear equations or to CSPs over structures with semilattice or majority polymorphism.