Computing the partition function for graph homomorphisms with multiplicities

TL;DR

This paper introduces a quasi-polynomial algorithm to compute a refined partition function for graph homomorphisms, enabling efficient differentiation between graphs with and without certain substructures.

Contribution

It presents a novel quasi-polynomial algorithm for the refined partition function, extending to various graph structures and coloring problems.

Findings

Efficient quasi-polynomial algorithms for partition functions of various graph structures.

Ability to distinguish graphs with different structural properties in quasi-polynomial time.

Applicable to problems like independent sets, matchings, Hamiltonian cycles, and colorings.

Abstract

We consider a refinement of the partition function of graph homomorphisms and present a quasi-polynomial algorithm to compute it in a certain domain. As a corollary, we obtain quasi-polynomial algorithms for computing partition functions for independent sets, perfect matchings, Hamiltonian cycles and dense subgraphs in graphs as well as for graph colorings. This allows us to tell apart in quasi-polynomial time graphs that are sufficiently far from having a structure of a given type (i.e., independent set of a given size, Hamiltonian cycle, etc.) from graphs that have sufficiently many structures of that type, even when the probability to hit such a structure at random is exponentially small.