# The Complexity of Ferromagnetic Two-spin Systems with External Fields

arXiv: https://arxiv.org/abs/1402.4346 · HTML: https://mlsanity.com/paper/1402.4346/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Feb 2014.

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

- [Rapid mixing in positively weighted restricted Boltzmann machines](https://mlsanity.com/paper/2604.00963/) (2026)

## Abstract (arXiv)

We study the approximability of computing the partition function for ferromagnetic two-state spin systems. The remarkable algorithm by Jerrum and Sinclair showed that there is a fully polynomial-time randomized approximation scheme (FPRAS) for the special ferromagnetic Ising model with any given uniform external field. Later, Goldberg and Jerrum proved that it is #BIS-hard for Ising model if we allow inconsistent external fields on different nodes. In contrast to these two results, we prove that for any ferromagnetic two-state spin systems except the Ising model, there exists a threshold for external fields beyond which the problem is #BIS-hard, even if the external field is uniform.