Polynomial-time approximation algorithms for the antiferromagnetic Ising model on line graphs

TL;DR

This paper introduces a polynomial-time Markov chain Monte Carlo algorithm for approximating the partition function of the antiferromagnetic Ising model on line graphs, leveraging winding technology, and demonstrates rapid mixing of Glauber dynamics on these graphs.

Contribution

It provides the first polynomial-time approximation algorithm for the antiferromagnetic Ising model on line graphs using winding technology.

Findings

The algorithm efficiently estimates the partition function.

Exact computation is #P-hard, so approximation is necessary.

Glauber dynamics are rapidly mixing on line graphs.

Abstract

We present a polynomial-time Markov chain Monte Carlo algorithm for estimating the partition function of the antiferromagnetic Ising model on any line graph. The analysis of the algorithm exploits the "winding" technology devised by McQuillan [CoRR abs/1301.2880 (2013)] and developed by Huang, Lu and Zhang [Proc. 27th Symp. on Disc. Algorithms (SODA16), 514-527]. We show that exact computation of the partition function is #P-hard, even for line graphs, indicating that an approximation algorithm is the best that can be expected. We also show that Glauber dynamics for the Ising model is rapidly mixing on line graphs, an example being the kagome lattice.