The complexity of the list homomorphism problem for graphs

TL;DR

This paper classifies the computational complexity of the list H-colouring problem for graphs, providing a complete algebraic and logical characterization that aligns with major conjectures in constraint satisfaction problems.

Contribution

It offers a complete classification of the problem's complexity for all graphs H, connecting combinatorial, algebraic, and logical perspectives.

Findings

Problems are either NP-complete, NL-complete, L-complete, or first-order definable.

Provides algebraic and descriptive complexity characterizations.

Aligns with key conjectures in constraint satisfaction theory.

Abstract

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.