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.
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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
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