Graph Homomorphisms and Universal Algebra

2026-02-15

AI Summary

This study investigates the computational complexity of constraint satisfaction problems (CSPs) over finite domains, aiming to precisely delineate the boundary between polynomial-time solvable and NP-hard cases. By adopting graph homomorphism as a unifying framework, the work systematically employs universal algebraic methods, with particular emphasis on the algebraic underpinnings of cyclic term operations and the bounded-width theorem. The project establishes a solvability criterion for CSPs based on algebraic invariants, which not only clarifies the theoretical foundations of existing complexity classifications but also provides an effective tool for identifying new tractable classes. This advances a deeper understanding of the fine-grained structure of the complexity landscape for finite-domain CSPs.

Technology Category

Constraint Satisfaction and Optimization: SatisfiabilityKnowledge Representation and Reasoning: Computational Complexity of ReasoningSearch and Optimization: Combinatorial Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• constraint satisfaction problems
• computational complexity
• polynomial-time tractability
• NP-hardness
• finite-domain CSPs

Innovation

• universal algebra
• constraint satisfaction problems
• graph homomorphisms
• cyclic terms
• bounded width