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 #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 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.Comment: Minor correction