Faster algorithms for graph homomorphism via tractable constraint satisfaction

TL;DR

This paper introduces faster algorithms for determining graph homomorphisms, especially for graphs excluding a topological minor, by reducing the problem to tractable constraint satisfaction problems, with improvements for specific cases like odd cycles.

Contribution

It provides a new algorithmic framework that achieves exponential time bounds for graph homomorphism problems on certain graph classes, extending previous methods.

Findings

Decides homomorphism existence in 2^{O(n)}h^{O(1)} time for graphs excluding a topological minor.

Reduces homomorphism problem to a single-exponential number of tractable CSPs.

Offers an improved randomized algorithm for homomorphisms to odd cycles.

Abstract

We show that the existence of a homomorphism from an $n$-vertex graph $G$ to an $h$-vertex graph $H$ can be decided in time $2^{O(n)}h^{O(1)}$ and polynomial space if $H$ comes from a family of graphs that excludes a topological minor. The algorithm is based on a reduction to a single-exponential number of constraint satisfaction problems over tractable languages and can handle cost minimization. We also present an improved randomized algorithm for the special case where the graph $H$ is an odd cycle.