Graph Homomorphisms and Universal Algebra

TL;DR

This paper introduces the universal-algebraic approach to analyzing the computational complexity of finite-domain constraint satisfaction problems, focusing on graph homomorphisms and related algebraic properties.

Contribution

It presents an accessible introduction to the universal-algebraic framework for CSPs, emphasizing cyclic terms and bounded width theorems in the context of graph homomorphisms.

Findings

Explains the connection between algebraic properties and CSP complexity.

Highlights the role of cyclic terms and bounded width in tractability.

Uses directed graphs and homomorphism problems as illustrative examples.

Abstract

Constraint satisfaction problems are computational problems that naturally appear in many areas of theoretical computer science. One of the central themes is their computational complexity, and in particular the border between polynomial-time tractability and NP-hardness. In this course we introduce the universal-algebraic approach to study the computational complexity of finite-domain CSPs. The course covers in particular the cyclic terms and bounded width theorems. To keep the presentation accessible, we start the course in the tangible setting of directed graphs and graph homomorphism problems.