Abstract
We provide evidence that it is computationally difficult to approximate the partition function of the ferromagnetic q-state Potts model when q > 2. Specifically we show that the partition function is hard for the complexity class #RHII1 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 bipartite graph. Our proof exploits the first order phase transition of the “random cluster” model, which is a probability distribution on graphs that is closely related to the q-state Potts model. A full version of this paper, with proofs included, is available at http://arxiv.org/abs/1002.0986 .
This work was partially supported by the EPSRC grant The Complexity of Counting in Constraint Satisfaction Problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alon, N., Frieze, A., Welsh, D.: Polynomial time randomized approximation schemes for Tutte-Gröthendieck invariants: the dense case. Random Structures Algorithms 6(4), 459–478 (1995)
Bollobás, B., Grimmett, G., Janson, S.: The random-cluster model on the complete graph. Probab. Theory Related Fields 104(3), 283–317 (1996)
Bordewich, M.: On the approximation complexity hierarchy (in preparation, 2010)
Chebolu, P., Goldberg, L.A., Martin, R.: Approximately counting stable matchings (in preparation, 2010)
Dalmau, V.: Linear datalog and bounded path duality of relational structures. Logical Methods in Computer Science 1(1) (2005)
Dyer, M.E., Goldberg, L.A., Greenhill, C.S., Jerrum, M.: The relative complexity of approximate counting problems. Algorithmica 38(3), 471–500 (2003)
Ge, Q., Stefankovic, D.: A graph polynomial for independent sets of bipartite graphs. CoRR, abs/0911.4732 (2009)
Goldberg, L.A., Jerrum, M.: Counterexample to rapid mixing of the GS Process. Technical note (2010)
Goldberg, L.A., Jerrum, M.: The complexity of ferromagnetic Ising with local fields. Combinatorics, Probability & Computing 16(1), 43–61 (2007)
Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Inform. and Comput. 206(7), 908–929 (2008)
Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial of a planar graph. CoRR, abs/0907.1724 (2009)
Grimmett, G.: Potts models and random-cluster processes with many-body interactions. J. Statist. Phys. 75(1-2), 67–121 (1994)
Holley, R.: Remarks on the FKG inequalities. Comm. Math. Phys. 36, 227–231 (1974)
Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108(1), 35–53 (1990)
Jerrum, M., Sinclair, A.: Polynomial-time approximation algorithms for the Ising model. SIAM J. Comput. 22(5), 1087–1116 (1993)
Jerrum, M.R., Valiant, L.G., Vazirani, V.V.: Random generation of combinatorial structures from a uniform distribution. Theoret. Comput. Sci. 43(2-3), 169–188 (1986)
Kelk, S.: On the relative complexity of approximately counting H-colourings. PhD thesis, University of Warwick, Coventry, UK (July 2004)
Luczak, M., Łuczak, T.: The phase transition in the cluster-scaled model of a random graph. Random Structures Algorithms 28(2), 215–246 (2006)
Potts, R.B.: Some generalized order-disorder transformations. Proc. Cambridge Philos. Soc. 48, 106–109 (1952)
Sokal, A.: The multivariate Tutte polynomial. In: Surveys in Combinatorics. Cambridge University Press, Cambridge (2005)
Vertigan, D.L., Welsh, D.J.A.: The computational complexity of the Tutte plane: the bipartite case. Combin. Probab. Comput. 1(2), 181–187 (1992)
Vertigan, D.: The computational complexity of Tutte invariants for planar graphs. SIAM J. Comput. 35(3), 690–712 (2005) (electronic)
Welsh, D.J.A.: Complexity: knots, colourings and counting. London Mathematical Society Lecture Note Series, vol. 186. Cambridge University Press, Cambridge (1993)
Zuckerman, D.: On unapproximable versions of NP-Complete problems. SIAM Journal on Computing 25(6), 1293–1304 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Goldberg, L.A., Jerrum, M. (2010). Approximating the Partition Function of the Ferromagnetic Potts Model. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_34
Download citation
DOI: https://doi.org/10.1007/978-3-642-14165-2_34
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14164-5
Online ISBN: 978-3-642-14165-2
eBook Packages: Computer ScienceComputer Science (R0)