Approximating the Partition Function of the Ferromagnetic Potts Model

摘要

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 #RHΠ_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 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.