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

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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 #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.
TitelGraph-Theoretic Concepts in Computer Science : 45th International Workshop, WG 2019, Vall de Núria, Spain, June 19–21, 2019, Revised Papers
ISBN (Trykt)978-3-030-30785-1
ISBN (Elektronisk)978-3-030-30786-8
StatusUdgivet - 2019
NavnLecture Notes in Computer Science


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