@inproceedings{91ddb7bcd6e04102936a80682b0242f0,
title = "The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms",
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 #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 {\v Z}ivn{\'y} (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 Lov{\'a}sz (Colloquium Publications 2012), this amounts to counting homomorphisms to quantum graphs.",
keywords = "Homomorphism counts, Complexity dichotomy, #P-hard, Exponential-time hypothesis, Quantum graphs, Homomorphism counts, Complexity dichotomy, #P-hard, Exponential-time hypothesis, Quantum graphs",
author = "Hubie Chen and Radu-Cristian Curticapean and Holger Dell",
year = "2019",
doi = "10.1007/978-3-030-30786-8_28",
language = "English",
isbn = "978-3-030-30785-1",
series = "Lecture Notes in Computer Science",
publisher = "Springer",
pages = "364--378",
booktitle = "Graph-Theoretic Concepts in Computer Science",
address = "Germany",
}