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We call an Ising model tractable when it is possible to compute its partitionfunction value (statistical inference) in polynomial time. The tractabilityalso implies an ability to sample configurations of this model in polynomialtime. The notion of tractability extends the basic case of planar zero-fieldIsing models. Our starting point is to describe algorithms for the basic casecomputing partition function and sampling efficiently. To derive thealgorithms, we use an equivalent linear transition to perfect matching countingand sampling on an expanded dual graph. Then, we extend our tractable inferenceand sampling algorithms to models, whose triconnected components are eitherplanar or graphs of $O(1)$ size. In particular, it results in a polynomial-timeinference and sampling algorithms for $K_{33}$ (minor) free topologies ofzero-field Ising models - a generalization of planar graphs with a potentiallyunbounded genus.

Inference and Sampling of $K_{33}$-free Ising Models