  An emerging trend in approximate counting is to show that certain
`low-temperature' problems are easy on typical instances, despite worst-case
hardness results. For the class of regular graphs one usually shows that
expansion can be exploited algorithmically, and since random regular graphs are
good expanders with high probability the problem is typically tractable.
Inspired by approaches used in subexponential-time algorithms for Unique Games,
we develop an approximation algorithm for the partition function of the
ferromagnetic Potts model on graphs with a small-set expansion condition. In
such graphs it may not suffice to explore the state space of the model close to
ground states, and a novel feature of our method is to efficiently find a
larger set of `pseudo-ground states' such that it is enough to explore the
model around each pseudo-ground state.
