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# A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

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LIPIcs.ICALP.2020.66.pdf
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## Cite As

Artem Govorov, Jin-Yi Cai, and Martin Dyer. A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 66:1-66:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.66

## Abstract

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 [Martin E. Dyer and Catherine S. Greenhill, 2000] 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 [Andrei Bulatov and Martin Grohe, 2005] 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. [Leslie A. Goldberg et al., 2010] for Z_A(⋅) also holds for simple graphs, where A is any real symmetric matrix.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Problems, reductions and completeness
##### Keywords
• Graph homomorphism
• Complexity dichotomy
• Counting problems

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