Approximate Counting via Correlation Decay on Planar Graphs

We show for a broad class of counting problems, correlation decay (strong spatial mixing) implies FPTAS on planar graphs. The framework for the counting problems considered by us is the Holant problems with arbitrary constant-size domain and symmetric constraint functions. We define a notion of regularity on the constraint functions, which covers a wide range of natural and important counting problems, including all multi-state spin systems, counting graph homomorphisms, counting weighted matchings or perfect matchings, the subgraphs world problem transformed from the ferromagnetic Ising model, and all counting CSPs and Holant problems with symmetric constraint functions of constant arity. The core of our algorithm is a fixed-parameter tractable algorithm which computes the exact values of the Holant problems with regular constraint functions on graphs of bounded treewidth. By utilizing the locally tree-like property of apex-minor-free families of graphs, the parameterized exact algorithm implies an FPTAS for the Holant problem on these graph families whenever the Gibbs measure defined by the problem exhibits strong spatial mixing. We further extend the recursive coupling technique to Holant problems and establish strong spatial mixing for the ferromagnetic Potts model and the subgraphs world problem. As consequences, we have new deterministic approximation algorithms on planar graphs and all apex-minor-free graphs for several counting problems.

💡 Research Summary

The paper studies a broad class of counting problems that can be expressed as Holant problems with arbitrary constant‑size domains and symmetric constraint functions. The authors introduce a notion of “regularity” for symmetric functions: a function is C‑regular if, after any pinning (fixing a subset of its arguments), the number of distinct resulting functions is bounded by a constant C. This definition captures many natural problems, including all multi‑state spin systems, graph homomorphisms, weighted matchings, perfect matchings, the subgraphs‑world transformation of the ferromagnetic Ising model, and any counting CSP or Holant problem with symmetric constraints of constant arity.

The technical core consists of two parts. First, they design a fixed‑parameter tractable (FPT) exact algorithm for Holant instances on graphs of bounded treewidth k. Instead of using a standard tree‑decomposition, they construct a “separator decomposition”: the graph is recursively split by small vertex separators into components of limited size. Conditioning on any assignment to a separator renders the sub‑components independent, which aligns with the conditional independence inherent in counting problems defined by local constraints. The algorithm runs in time 2^{O(k)}·poly(n), where n is the number of vertices.

Second, they show that for apex‑minor‑free graph families (which include planar graphs), strong spatial mixing (SSM) of the associated Gibbs measure implies a deterministic fully polynomial‑time approximation scheme (FPTAS) for the Holant problem. The key observation is that SSM on the original graph guarantees that distant variables have negligible influence on each other, allowing the original graph to be approximated by a collection of bounded‑treewidth pieces without any exponential blow‑up. Consequently, the “decay‑only’’ results from correlation‑decay literature can be directly turned into an FPTAS, unlike earlier approaches that required constructing large self‑avoiding‑walk trees.

To establish SSM for specific problems, the authors extend the recursive coupling technique of Goldberg, Jerrum, and Paterson to the Holant setting. They prove SSM for the ferromagnetic Potts model and for the subgraphs‑world problem. For the Potts model they obtain a mixing condition β < ln((q‑2)Δ‑1)/(Δ+1) on planar graphs of maximum degree Δ, which significantly improves previous bounds and approaches the conjectured O(1/Δ) threshold. For the subgraphs‑world (and thus the ferromagnetic Ising model with arbitrary external field) they give explicit degree constraints involving the parameters μ and λ.

As concrete algorithmic consequences, the paper yields deterministic FPTASes on planar and, more generally, all apex‑minor‑free graphs for:

• Counting q‑colorings on triangle‑free planar graphs when q > αΔ – γ (with α≈1.7632, γ≈0.47031);
• The subgraphs‑world model with μ, λ < 1 under a degree bound Δ < (1+λμ²)²/(1–μ²);
• The ferromagnetic q‑state Potts model under the aforementioned β bound.

Overall, the work bridges exact parameterized computation and approximation via correlation decay, providing a unified framework that handles a wide variety of counting problems beyond binary spin systems. It demonstrates that regularity of constraint functions, together with separator‑based exact algorithms, suffices to translate strong spatial mixing on the original graph into deterministic approximation schemes for complex counting problems on planar and related sparse graph families.