Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case

TL;DR

This paper classifies the computational complexity of certain symmetric promise constraint satisfaction problems involving hypergraph colorings, extending understanding beyond Boolean cases and identifying an open problem.

Contribution

It provides an almost complete complexity classification for symmetric PCSPs on 3-uniform hypergraphs, advancing the theoretical understanding of these problems.

Findings

Most PCSPs of this form are classified as either polynomial-time solvable or NP-hard.

The paper identifies a specific open problem related to a hypergraph coloring with a particular relation.

It extends the complexity theory of PCSPs beyond Boolean and simple graph cases.

Abstract

The Promise Constraint Satisfaction Problem (PCSP) is a recently introduced vast generalization of the Constraint Satisfaction Problem (CSP). We investigate the computational complexity of a class of PCSPs beyond the most studied cases - approximation variants of satisfiability and graph coloring problems. We give an almost complete classification for the class of PCSPs of the form: given a 3-uniform hypergraph that has an admissible 2-coloring, find an admissible 3-coloring, where admissibility is given by a ternary symmetric relation. The only PCSP of this sort whose complexity is left open in this work is a natural hypergraph coloring problem, where admissibility is given by the relation "if two colors are equal, then the remaining one is higher."