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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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.

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)

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@InProceedings{govorov_et_al:LIPIcs.ICALP.2020.66, author = {Govorov, Artem and Cai, Jin-Yi and Dyer, Martin}, title = {{A Dichotomy for Bounded Degree Graph Homomorphisms with Nonnegative Weights}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {66:1--66:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.66}, URN = {urn:nbn:de:0030-drops-124733}, doi = {10.4230/LIPIcs.ICALP.2020.66}, annote = {Keywords: Graph homomorphism, Complexity dichotomy, Counting problems} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Graph homomorphism has been an important research topic since its introduction [László Lovász, 1967]. Stated in the language of binary relational structures in that paper [László Lovász, 1967], Lovász proved a fundamental theorem that the graph homomorphism function G ↦ hom(G, H) for 0-1 valued H (as the adjacency matrix of a graph) determines the isomorphism type of H. In the past 50 years various extensions have been proved by Lovász and others [László Lovász, 2006; Michael Freedman et al., 2007; Christian Borgs et al., 2008; Alexander Schrijver, 2009; László Lovász and Balázs Szegedy, 2009]. These extend the basic 0-1 case to admit vertex and edge weights; but always with some restrictions such as all vertex weights must be positive. In this paper we prove a general form of this theorem where H can have arbitrary vertex and edge weights. An innovative aspect is that we prove this by a surprisingly simple and unified argument. This bypasses various technical obstacles and unifies and extends all previous known versions of this theorem on graphs. The constructive proof of our theorem can be used to make various complexity dichotomy theorems for graph homomorphism effective, i.e., it provides an algorithm that for any H either outputs a P-time algorithm solving hom(⋅, H) or a P-time reduction from a canonical #P-hard problem to hom(⋅, H).

Jin-Yi Cai and Artem Govorov. On a Theorem of Lovász that hom(⋅, H) Determines the Isomorphism Type of H. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2020.17, author = {Cai, Jin-Yi and Govorov, Artem}, title = {{On a Theorem of Lov\'{a}sz that hom(⋅, H) Determines the Isomorphism Type of H}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {17:1--17:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.17}, URN = {urn:nbn:de:0030-drops-117022}, doi = {10.4230/LIPIcs.ITCS.2020.17}, annote = {Keywords: Graph homomorphism, Partition function, Complexity dichotomy, Connection matrices and tensors} }

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