  We investigate the computational difficulty of approximating the partition
function of the ferromagnetic Ising model on a regular matroid. Jerrum and
Sinclair have shown that there is a fully polynomial randomised approximation
scheme (FPRAS) for the class of graphic matroids. On the other hand, the
authors have previously shown, subject to a complexity-theoretic assumption,
that there is no FPRAS for the class of binary matroids, which is a proper
superset of the class of graphic matroids. In order to map out the region where
approximation is feasible, we focus on the class of regular matroids, an
important class of matroids which properly includes the class of graphic
matroids, and is properly included in the class of binary matroids. Using
Seymour's decomposition theorem, we give an FPRAS for the class of regular
matroids.
