Constraint Satisfaction Problems over Finite Structures

TL;DR

This paper explores the computational complexity of CSPs over finite structures with relations and operations, linking it to algebraic properties of finite algebras and extending existing classifications.

Contribution

It introduces a novel algebraic perspective on CSP complexity, providing conditions for finite algebras and classifying two-element structures with new insights.

Findings

Finite equationally nontrivial algebras admit only polynomially many homomorphisms.

Complete classification of CSPs over two-element structures extended.

Examples of structures with bounded width but not relational width.

Abstract

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a natural algebraic question: which finite algebras admit only polynomially many homomorphisms into them? We give some sufficient and some necessary conditions for a finite algebra to have this property. In particular, we show that every finite equationally nontrivial algebra has this property which gives us, as a simple consequence, a complete complexity classification of CSPs over two-element structures, thus extending the classification for two-element relational structures by Schaefer (STOC'78). We also present examples of two-element structures that have bounded width but do not have relational width (2,3), thus demonstrating that, from a descriptive complexity perspective, allowing operations leads to a richer theory.