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International Workshop on Graph-Theoretic Concepts in Computer Science

WG 2019: Graph-Theoretic Concepts in Computer Science pp 364–378Cite as

The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11789))

Abstract

Many graph parameters can be expressed as homomorphism counts to fixed target graphs; this includes the number of independent sets and the number of k-colorings for any fixed k. Dyer and Greenhill (RSA 2000) gave a sweeping complexity dichotomy for such problems, classifying which target graphs render the problem polynomial-time solvable or \(\#\mathrm {P}\)-hard. In this paper, we give a new and shorter proof of this theorem, with previously unknown tight lower bounds under the exponential-time hypothesis. We similarly strengthen complexity dichotomies by Focke, Goldberg, and Živný (SODA 2018) for counting surjective homomorphisms to fixed graphs. Both results crucially rely on our main contribution, a complexity dichotomy for evaluating linear combinations of homomorphism numbers to fixed graphs. In the terminology of Lovász (Colloquium Publications 2012), this amounts to counting homomorphisms to quantum graphs.

Hubie Chen acknowledges the support of Spanish Project TIN2017-86727-C2-2-R. Radu Curticapean was partly supported by ERC grant SYSTEMATICGRAPH (No. 725978) and VILLUM Foundation grant 16582 while working on this project.

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Author information

Authors and Affiliations

  1. Department of Computer Science and Information Systems, Birkbeck University of London, London, UK

    Hubie Chen

  2. Basic Algorithms Research Copenhagen (BARC), Copenhagen, Denmark

    Radu Curticapean

  3. IT University of Copenhagen, Copenhagen, Denmark

    Radu Curticapean & Holger Dell

Authors
  1. Hubie Chen
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  2. Radu Curticapean
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  3. Holger Dell
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Corresponding author

Correspondence to Radu Curticapean .

Editor information

Editors and Affiliations

  1. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Ignasi Sau

  2. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Dimitrios M. Thilikos

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Chen, H., Curticapean, R., Dell, H. (2019). The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_28

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  • DOI: https://doi.org/10.1007/978-3-030-30786-8_28

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  • Online ISBN: 978-3-030-30786-8

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