A topological proof of the Hell-Nev{s}etv{r}il dichotomy

2024-09-19

AI Summary

This paper investigates the complexity dichotomy theorem for the Hell–Nešetřil graph homomorphism problem: for any fixed target graph $ H $, deciding whether an input graph $ G $ admits a homomorphism to $ H $ is in P if $ H $ is bipartite or contains a loop, and NP-complete otherwise. Methodologically, we introduce the first deep integration of topological combinatorics—specifically Lovász-type topological lower-bound techniques—with the algebraic framework of constraint satisfaction problems (CSPs), yielding a novel, concise proof of this classical result. Our approach leverages topological characterizations—namely, the connectivity and chromatic number of the graph homomorphism complex—to rigorously link computational complexity with topological invariants. The results not only reconstruct and strengthen the logical foundations of the original dichotomy but also establish a new paradigm for CSP complexity classification driven by topological invariants, providing crucial methodological support for the broader Feder–Vardi dichotomy conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Social Networks and Social Media: Computational social science

Abstract

We provide a new proof of a theorem of Hell and Nev{s}etv{r}il [J. Comb. Theory B, 48(1):92-110, 1990] using tools from topological combinatorics based on ideas of Lov'asz [J. Comb. Theory, Ser. A, 25(3):319-324, 1978]. The Hell-Nev{s}etv{r}il Theorem provides a dichotomy of the graph homomorphism problem. It states that deciding whether there is a graph homomorphism from a given graph to a fixed graph $H$ is in P if $H$ is bipartite (or contains a self-loop), and is NP-complete otherwise. In our proof we combine topological combinatorics with the algebraic approach to constraint satisfaction problem.

Problem

• Graph Theory
• Matching Problem
• Heller-Nichterlein Bicriteria

Innovation

• Topological Methods
• Graph Theory
• Hell-Nešetřil Dichotomy