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

TL;DR

This paper introduces a new class of zero-field Ising models that are efficiently solvable by dynamic programming, extending tractability to certain non-planar topologies like $K_{3,3}$- and $K_5$-minor-free graphs, and improves inference approximation in grid models.

Contribution

It develops a polynomial-time inference and sampling algorithm for minor-free zero-field Ising models, expanding tractability beyond planar graphs.

Findings

Efficient inference and sampling for $K_{3,3}$- and $K_5$-minor-free topologies.

Extension of tractable models to non-genus, non-treewidth bounded graphs.

Empirical improvement in inference approximation for grid Ising models with magnetic fields.

Abstract

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 $K_{3,3}$-minor-free topologies and over $K_{5}$-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.