A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory
Suppose \varphi and \psi are two angles satisfying \tan(\varphi) = 2 \tan(\psi) > 0. We prove that under this condition \varphi and \psi cannot be both rational multiples of \pi. We use this number theoretic result to prove a classification of the computational complexity of spin systems on k-regular graphs with general (not necessarily symmetric) real valued edge weights. We establish explicit criteria, according to which the partition functions of all such systems are classified into three classes: (1) Polynomial time
computable, (2) \#P-hard in general but polynomial time computable
on planar graphs, and (3) \#P-hard on planar graphs. In particular problems in (2) are precisely those that can be transformed to a form solvable by the Fisher-Kasteleyn-Temperley algorithm by a holographic reduction.
Spin Systems
Holant Problems
Number Theory
Characters
Cyclotomic Fields
2:1-2:22
Regular Paper
Jin-Yi
Cai
Jin-Yi Cai
Zhiguo
Fu
Zhiguo Fu
Kurt
Girstmair
Kurt Girstmair
Michael
Kowalczyk
Michael Kowalczyk
10.4230/LIPIcs.ITCS.2018.2
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode