The antiferromagnetic Ising model beyond line graphs

TL;DR

This paper investigates the computational complexity of the antiferromagnetic Ising model beyond line graphs, revealing limitations in extending tractability results known for related models like the hard-core model.

Contribution

It demonstrates that the tractability extensions applicable to the hard-core model do not carry over to the antiferromagnetic Ising model beyond line graphs.

Findings

Tractability of the antiferromagnetic Ising model does not extend beyond line graphs.

Hard-core model tractability extends to claw-free graphs, unlike the Ising model.

Shows fundamental differences in complexity between the two models.

Abstract

Both the antiferromagnetic Ising model and the hard-core model could be said to be tractable on line graphs of bounded degree. For example, Glauber dynamics is rapidly mixing in both cases. In the case of the hard-core model, we know that tractability extends further, to claw-free graphs and somewhat beyond. In contrast, it is shown here that the corresponding extensions are not possible in the case of the antiferromagnetic Ising model.