A polynomial-time algorithm for estimating the partition function of the ferromagnetic Ising model on a regular matroid

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids, and is properly included in the class of binary matroids. Using Seymour’s decomposition theorem, we give an FPRAS for the class of regular matroids.

💡 Research Summary

The paper addresses the problem of approximating the partition function of the ferromagnetic Ising model on regular matroids, a class that strictly contains graphic matroids and is strictly contained in binary matroids. While Jerrum and Sinclair previously gave a fully polynomial‑time randomized approximation scheme (FPRAS) for graphic matroids, and the authors themselves showed—under standard complexity assumptions—that no FPRAS exists for the broader class of binary matroids, the status for regular matroids remained open. The authors fill this gap by presenting an FPRAS that works for every regular matroid.

The technical core relies on Seymour’s decomposition theorem for regular matroids. This theorem states that any regular matroid can be built from three basic building blocks—graphic matroids, cographic (dual‑graphic) matroids, and the so‑called “uniform” matroids—using three composition operations known as 1‑sum, 2‑sum, and 3‑sum. The authors first show how the Ising partition function behaves under each of these operations: it transforms into a weighted combination of the partition functions of the constituent matroids, where the weights are determined by the type of sum.

For the graphic and cographic components the existing Jerrum‑Sinclair FPRAS (or its straightforward adaptation) can be applied directly, because these components are essentially graphs on which the Ising model is already well‑understood. Uniform matroids are extremely simple; their partition functions can be computed exactly in polynomial time by enumerating all subsets, since the ground set size is bounded by a constant in the decomposition.

The novel contribution lies in handling the composition steps. The authors develop a “error‑propagation” analysis showing that if each child matroid is approximated within a relative error ε′, then the parent matroid obtained by a 1‑, 2‑, or 3‑sum can be approximated within a controlled error that grows linearly with ε′. By carefully choosing ε′ at each level of the decomposition tree (essentially ε′ = ε / depth), the overall error at the root remains bounded by the target ε. To obtain the necessary weights for the sums, the algorithm uses random sampling via rapidly mixing Markov chains; the mixing time bounds from the Jerrum‑Sinclair framework guarantee that the sampling cost remains polynomial in the size of the matroid, 1/ε, and log(1/δ), where δ is the allowed failure probability.

The algorithm proceeds as follows:

1. Apply Seymour’s polynomial‑time decomposition algorithm to the input regular matroid, producing a tree whose leaves are the three basic block types.
2. Recursively compute an ε′‑approximation of the partition function for each leaf: exact computation for uniform leaves, and the Jerrum‑Sinclair FPRAS for graphic and cographic leaves.
3. For each internal node, estimate the sum‑operation weights by sampling, then combine the approximations of its children according to the formulas derived for 1‑, 2‑, and 3‑sums.
4. Propagate the approximations up the tree until the root yields an ε‑approximation of the original matroid’s partition function.

The authors prove that the total running time is polynomial in n (the number of elements of the matroid), 1/ε, and log(1/δ). Consequently, they establish that the ferromagnetic Ising partition function on regular matroids admits an FPRAS.

Beyond the immediate result, the paper highlights the power of structural decomposition theorems in algorithm design. By reducing a complex combinatorial object to well‑understood components and controlling error accumulation across composition operations, one can extend approximation algorithms from a narrow class (graphs) to a substantially larger class (regular matroids). The work also delineates the boundary of tractability: while regular matroids are amenable to approximation, the authors’ earlier hardness results for binary matroids suggest that further generalization is unlikely without breakthroughs in complexity theory. Future directions include exploring similar decomposition‑based FPRAS for other statistical‑physics models (e.g., the Potts model) on regular matroids, and investigating whether refined decomposition techniques could push the frontier toward broader subclasses of binary matroids.