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A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights

Authors Artem Govorov, Jin-Yi Cai, Martin Dyer

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

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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.

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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

Author Details

Artem Govorov
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA
Jin-Yi Cai
  • Department of Computer Sciences, University of Wisconsin-Madison, Madison, WI, USA
Martin Dyer
  • School of Computing, University of Leeds, UK

Funding

  • Govorov, Artem: Supported by NSF CCF-1714275.
  • Cai, Jin-Yi: Supported by NSF CCF-1714275.
  • Dyer, Martin: Supported by EPSRC grant EP/S016562/1.

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