Computing the partition function for graph homomorphisms

TL;DR

This paper introduces a new partition function for edge-colored graph homomorphisms, providing an efficient approximation algorithm with applications to counting colorings, independent sets, and graph isomorphism testing.

Contribution

It presents the first efficient approximation algorithm for the partition function of edge-colored graph homomorphisms, extending previous work on uncolored homomorphisms.

Findings

Efficient approximation algorithm for the partition function.

Applications to counting k-colorings and independent sets.

Procedure to distinguish graph pairs with many homomorphisms.

Abstract

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include efficient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an efficient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G --> H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G --> H.