# Tractable Minor-free Generalization of Planar Zero-field Ising Models

arXiv: https://arxiv.org/abs/1910.11142 · HTML: https://mlsanity.com/paper/1910.11142/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Oct 2019.

## Built on by (most cited first, 1 of 1)

- [Inference and Optimization for Engineering and Physical Systems](https://mlsanity.com/paper/2208.13880/) (2022)

## Abstract (arXiv)

We present a new family of zero-field Ising models over N binary variables/spins obtained by consecutive "gluing" of planar and O(1)-sized components and subsets of at most three vertices into a tree. The polynomial-time algorithm of the dynamic programming type for solving exact inference (computing partition function) and exact sampling (generating i.i.d. samples) consists in a sequential application of an efficient (for planar) or brute-force (for O(1)-sized) inference and sampling to the components as a black box. To illustrate the utility of the new family of tractable graphical models, we first build a polynomial algorithm for inference and sampling of zero-field Ising models over K3,3-minor-free topologies and over K5-minor-free topologies -- both are extensions of the planar zero-field Ising models -- which are neither genus - nor treewidth-bounded. Second, we demonstrate empirically an improvement in the approximation quality of the NP-hard problem of inference over the square-grid Ising model in a node-dependent non-zero "magnetic" field.