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DOI: 10.4230/LIPIcs.ITCS.2018.2
A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory

Authors: Jin-Yi Cai, Zhiguo Fu, Kurt Girstmair, and Michael Kowalczyk

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Abstract
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.
Cite as

Jin-Yi Cai, Zhiguo Fu, Kurt Girstmair, and Michael Kowalczyk. A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 2:1-2:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2018.2,
  author =	{Cai, Jin-Yi and Fu, Zhiguo and Girstmair, Kurt and Kowalczyk, Michael},
  title =	{{A Complexity Trichotomy for k-Regular Asymmetric Spin Systems Using Number Theory}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{2:1--2:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.2},
  URN =		{urn:nbn:de:0030-drops-83251},
  doi =		{10.4230/LIPIcs.ITCS.2018.2},
  annote =	{Keywords: Spin Systems, Holant Problems, Number Theory, Characters, Cyclotomic Fields}
}
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