eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-01-12
2:1
2:22
10.4230/LIPIcs.ITCS.2018.2
article
A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory
Cai, Jin-Yi
Fu, Zhiguo
Girstmair, Kurt
Kowalczyk, Michael
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol094-itcs2018/LIPIcs.ITCS.2018.2/LIPIcs.ITCS.2018.2.pdf
Spin Systems
Holant Problems
Number Theory
Characters
Cyclotomic Fields