We give algorithms for approximating the partition function of the
ferromagnetic Potts model on d-regular expanding graphs. We require much
weaker expansion than in previous works; for example, the expansion exhibited
by the hypercube suffices. The main improvements come from a significantly
sharper analysis of standard polymer models, using extremal graph theory and
applications of Karger's algorithm to counting cuts that may be of independent
interest. It is #BIS-hard to approximate the partition function at low
temperatures on bounded-degree graphs, so our algorithm can be seen as evidence
that hard instances of #BIS are rare. We believe that these methods can shed
more light on other important problems such as sub-exponential algorithms for
approximate counting problems.Comment: 27 page