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Complexity of Ising Polynomials

Published online by Cambridge University Press:  04 July 2012

TOMER KOTEK
TOMER KOTEK*
Affiliation:
Department of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel (e-mail: [email protected])
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Abstract

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by Andrén and Markström.

We consider the complexity of Z(Gt,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show that Z(G;x,y,z) is #P-hard to evaluate at all points in 3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETH a dichotomy theorem stating that evaluations of Z(G;t,y) either take exponential time in the number of vertices of G to compute, or can be done in polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

Keywords

Primary 05C31Secondary 05C8582B20
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 5 , September 2012 , pp. 743 - 772
Copyright
Copyright © Cambridge University Press 2012

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