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

## TL;DR

This paper introduces polynomial-time algorithms for inference and sampling in a broad class of Ising models, including those with $K_{33}$-free topologies, extending beyond planar graphs.

## Contribution

It extends tractable inference and sampling algorithms to $K_{33}$-free Ising models, generalizing planar cases to models with complex topologies.

## Key findings

- Polynomial-time algorithms for $K_{33}$-free Ising models.
- Extension of tractability from planar to $K_{33}$-free topologies.
- Efficient sampling and inference in models with unbounded genus.

## Abstract

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