Govorov, A, Cai, JY and Dyer, M orcid.org/0000-0002-2018-0374 (2020) A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights. In: LIPIcs : Leibniz International Proceedings in Informatics. 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 08-11 Jul 2020, Online. Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik , 66:1-66:18. ISBN 9783959771382
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.
Metadata
Item Type: | Proceedings Paper |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © Artem Govorov, Jin-Yi Cai, and Martin Dyer. This is an open access article under the terms of the Creative Commons Attribution 3.0 Unported (CC BY 3.0) (https://creativecommons.org/licenses/by/3.0/) |
Keywords: | Graph homomorphism; Complexity dichotomy; Counting problems |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) |
Funding Information: | Funder Grant number EPSRC (Engineering and Physical Sciences Research Council) EP/S016562/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 18 Sep 2020 10:23 |
Last Modified: | 02 Nov 2021 09:35 |
Status: | Published |
Publisher: | Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik |
Identification Number: | 10.4230/LIPIcs.ICALP.2020.66 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:165280 |