Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

TL;DR

This paper extends the algebraic framework used to classify the complexity of constraint satisfaction problems (CSPs) to a broader class called promise CSPs, which include relaxed and approximate problems, revealing their underlying symmetry-based complexity structure.

Contribution

It introduces an algebraic theory for promise CSPs, broadening the scope of existing complexity classifications for standard CSPs.

Findings

Complexity of promise CSPs is characterized by specific symmetries.

The algebraic approach applies to relaxed and approximate problems.

Provides foundational steps towards a full complexity classification for promise CSPs.

Abstract

What makes a computational problem easy (e.g., in P, that is, solvable in polynomial time) or hard (e.g., NP-hard)? This fundamental question now has a satisfactory answer for a quite broad class of computational problems, so called fixed-template constraint satisfaction problems (CSPs) -- it has turned out that their complexity is captured by a certain specific form of symmetry. This paper explains an extension of this theory to a much broader class of computational problems, the promise CSPs, which includes relaxed versions of CSPs such as the problem of finding a 137-coloring of a 3-colorable graph.