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The complexity of approximating complex-valued Ising and Tutte partition functions [article]

Leslie Ann Goldberg, Heng Guo
2017 arXiv   pre-print
We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters.  ...  The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters.  ...  Acknowledgements We thank Dan Shepherd and Mark Jerrum for useful discussions.  ... 
arXiv:1409.5627v4 fatcat:vzlt7m7gx5cu7k45gnabaxvjoq

The Complexity of Approximating complex-valued Ising and Tutte partition functions

Leslie Ann Goldberg, Heng Guo
2017 Computational Complexity  
We study the complexity of approximately evaluating the Ising and Tutte partition functions with complex parameters.  ...  The motivation for this paper is to study more comprehensively the complexity of (classically) approximating the Ising and Tutte partition functions with complex parameters.  ...  BQP and the Tutte polynomial Bordewich et al. [4] raised the question "of determining whether the Tutte polynomial is greater than or equal to, or less than zero at a given point."  ... 
doi:10.1007/s00037-017-0162-2 fatcat:whewokm6sbdqhpzsn47bwpym2e

Approximating the partition function of the ferromagnetic potts model

Leslie Ann Goldberg, Mark Jerrum
2012 Journal of the ACM  
Specifically we show that the partition function is hard for the complexity class #RHPi_1 under approximation-preserving reducibility.  ...  Thus, it is as hard to approximate the partition function as it is to find approximate solutions to a wide range of counting problems, including that of determining the number of independent sets in a  ...  As formulated in (2) , the Tutte polynomial is the partition function of this model.  ... 
doi:10.1145/2371656.2371660 fatcat:5o5ql5h3bvez3k6vwcxyrtrtcq

A little statistical mechanics for the graph theorist

Laura Beaudin, Joanna Ellis-Monaghan, Greta Pangborn, Robert Shrock
2010 Discrete Mathematics  
We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions.  ...  We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy  ...  What is the relationship between the Potts model and the chromatic polynomial? What is the computational complexity of the partition function?  ... 
doi:10.1016/j.disc.2010.03.011 fatcat:roiu7evabzg3bojnbvhl56ebdm

The Tutte–Potts connection in the presence of an external magnetic field

Joanna A. Ellis-Monaghan, Iain Moffatt
2011 Advances in Applied Mathematics  
We prove that the variable field Potts model partition function (with its many specializations) is an evaluation of the V-polynomial, and hence a polynomial with deletion-contraction reduction and Fortuin-Kasteleyn  ...  The classical relationship between the Tutte polynomial of graph theory and the Potts model of statistical mechanics has resulted in valuable interactions between the disciplines.  ...  Acknowledgments We thank Alain Brizard, Leslie Ann Goldberg, Maria Kiskowski, Steve Noble, Robert Shrock, Jim Stasheff, and especially Alan Sokal, for several informative conversations.  ... 
doi:10.1016/j.aam.2011.02.004 fatcat:qvbj4cnjxnehdiwq4ogudltvpa

Deterministic Polynomial-Time Approximation Algorithms for Partition Functions and Graph Polynomials

Viresh Patel, Guus Regts
2017 SIAM journal on computing (Print)  
In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models.  ...  More specifically, we define a large class of graph polynomials C and show that if p∈ C and there is a disk D centered at zero in the complex plane such that p(G) does not vanish on D for all bounded degree  ...  Acknowledgements We thank Alexander Barvinok for stimulating discussions, useful remarks and for sharing the results in [6] with us. We thank Andreas Galanis for informing us about [42] .  ... 
doi:10.1137/16m1101003 fatcat:3eur4cw3rfdtvcesbo55vc6kjm

The Complexity of Approximating the Complex-Valued Potts Model

Andreas Galanis, Leslie Ann Goldberg, Andrés Herrera-Poyatos, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science  
We study the complexity of approximating the partition function of the q-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters.  ...  precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters.  ...  Hence, we can translate our results on the complexity of approximating the Tutte polynomial of a planar graph to the complexity of approximating the Jones polynomial of an alternating link, and obtain  ... 
doi:10.4230/lipics.mfcs.2020.36 dblp:conf/mfcs/GalanisGH20 fatcat:ux7zybhsxvgezohy5puqoseeem

Stability-to-instability transition in the structure of large-scale networks

Dandan Hu, Peter Ronhovde, Zohar Nussinov
2012 Physical Review E  
Given the relation between Tutte and Jones polynomials, our results further suggest a link between the above complexity transitions and transitions associated with random knots.  ...  We examine phase transitions between the easy, hard, and the unsolvable phases when attempting to identify structure in large complex networks (community detection) in the presence of disorder induced  ...  Tran, and L. Zdeborová for discussions and ongoing work.  ... 
doi:10.1103/physreve.86.066106 pmid:23368003 fatcat:aarsvkj55nhuzpk77fsjarjfx4

Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials

Leslie Ann Goldberg, Mark Jerrum
2013 Journal of computer and system sciences (Print)  
We consider the problem of approximating certain combinatorial polynomials. First, we consider the problem of approximating the Tutte polynomial of a binary matroid with parameters q>= 2 and gamma.  ...  For binary matroids, we extend this result by showing (i) there is no FPRAS in the region gamma<-2 unless NP=RP, and (ii) in the region gamma>0, the approximation problem is hard for the complexity class  ...  (5) and (6) in the case q = 2 to see how they both arise as natural generalisations of the Ising partition function of a graph.  ... 
doi:10.1016/j.jcss.2012.04.005 fatcat:xtdnriwx6bdavpaoft64sk3cvy

The number of link and cluster states: the core of the 2Dqstate Potts model

J Hove
2005 Journal of Physics A: Mathematical and General  
A key element of the Random Cluster representation is the combinatorial factor Γ_G(,), which is the number of ways to form distinct clusters, consisting of totally edges.  ...  Due to Fortuin and Kastelyin the q state Potts model has a representation as a sum over random graphs, generalizing the Potts model to arbitrary q is based on this representation.  ...  Zeros in the complex q plane The formulation of the partition function as a polynomial in q allows for quite easy evaluation of the zeros of the partition function in the complex q plane.  ... 
doi:10.1088/0305-4470/38/50/002 fatcat:rayvb4gjrfgnbissm3dliop3n4

Quantum Computation and the Evaluation of Tensor Networks

Itai Arad, Zeph Landau
2010 SIAM journal on computing (Print)  
We use this algorithm to derive new quantum algorithms that approximate the partition function of a variety of classical statistical mechanics models, including the Potts model.  ...  We present a quantum algorithm that additively approximates the value of a tensor network to a certain scale.  ...  Acknowledgments We would like to thank Maarten van den Nest, Wolfgang Dür and Hans Briegel for pointing our attention to an implicit quantum algorithm for the evaluation of the partition function of the  ... 
doi:10.1137/080739379 fatcat:ewqkttj5t5cnzbdspmu5vsznly

Inapproximability of the Tutte polynomial of a planar graph

Leslie Ann Goldberg, Mark Jerrum
2012 Computational Complexity  
Vertigan completely mapped the complexity of exactly computing the Tutte polynomial of a planar graph.  ...  We study the complexity of the following problem, for rationals x and y: given as input a planar graph G, determine T(G;x,y).  ...  The hyperbola H 2 is of particular interest, as the Tutte polynomial here corresponds to the partition function of the celebrated Ising model in statistical physics.  ... 
doi:10.1007/s00037-012-0046-4 fatcat:cw3pgnkrezemfk32fpyce4xuh4

On the algebraic complexity of some families of coloured Tutte polynomials

Martin Lotz, Johann A. Makowsky
2004 Advances in Applied Mathematics  
Generalising the well-known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynomial, thus  ...  We investigate the coloured Tutte polynomial in Valiant's algebraic framework of NP-completeness.  ...  We also thank the anonymous referee for some useful comments and suggestions.  ... 
doi:10.1016/s0196-8858(03)00087-3 fatcat:4ui5xgsqzffilkh4valptjqr74

On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers

Joseph Geraci, Daniel A. Lidar
2008 Communications in Mathematical Physics  
This problem is related to the evaluation of the Jones and Tutte polynomials.  ...  We present an efficient quantum algorithm for the exact evaluation of either the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function Z for a family of graphs related to irreducible  ...  (9) and complexity results concerning the Tutte polynomial, that the evaluation of the Potts partition function is also ♯P-hard.  ... 
doi:10.1007/s00220-008-0438-0 fatcat:y5o7g27mr5et3h5nfhxxpziiwu

New graph polynomials from the Bethe approximation of the Ising partition function [article]

Yusuke Watanabe, Kenji Fukumizu
2010 arXiv   pre-print
One is a polynomial of two variables whose investigation is motivated by the performance analysis of the Bethe approximation of the Ising partition function.  ...  It is shown that these polynomials satisfy deletion-contraction relations and are new examples of the V-function, which was introduced by Tutte (1947, Proc. Cambridge Philos. Soc. 43, 26-40).  ...  Acknowledgments This work was supported in part by Grant-in-Aid for JSPS Fellows 20-993 and Grant-in-Aid for Scientific Research (C) 19500249.  ... 
arXiv:0908.3850v2 fatcat:wfqimmothvdebdxxouqz4cwsrm
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