A dichotomy for bounded degree graph homomorphisms with nonnegative weights

Abstract

Each symmetric matrix A defines a graph homomorphism function ZA(⋅), also known as the partition function. We prove that the Bulatov-Grohe dichotomy [4] for ZA(⋅) holds for bounded degree graphs. This resolves a problem that has been open for 15 years. Specifically, we prove that for any nonnegative symmetric matrix A with algebraic entries, either ZA(G) is in polynomial time for all graphs G, or it is #P-hard for bounded degree (and simple) graphs G. We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [12] for ZA(⋅) also holds for simple graphs, where A is any real symmetric matrix.