Polynomial-time Approximation Algorithms for the Antiferromagnetic Ising Model on Line Graphs.

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