A dichotomy for bounded degree graph homomorphisms with nonnegative weights
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We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A. Each symmetric matrix A defines a graph homomorphism function Z_A(·), also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of Z_A(·) for symmetric {0, 1}-matrices A, and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [4] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A. However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A, either Z_A(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 Z_A(·) also holds for simple graphs, where A is any real symmetric matrix.
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