TY - GEN

T1 - The Exponential-Time Complexity of Counting (Quantum) Graph Homomorphisms

AU - Chen, Hubie

AU - Curticapean, Radu-Cristian

AU - Dell, Holger

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

U2 - 10.1007/978-3-030-30786-8_28

DO - 10.1007/978-3-030-30786-8_28

M3 - Article in proceedings

SN - 978-3-030-30785-1

T3 - Lecture Notes in Computer Science

SP - 364

EP - 378

BT - Graph-Theoretic Concepts in Computer Science

PB - Springer

ER -