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 establishes a broad and unified framework for deterministic approximation of a large class of counting problems on planar and, more generally, apex‑minor‑free graph families. The authors work within the Holant formalism, where variables take values from a constant‑size domain and constraints are represented by symmetric functions. They introduce the notion of “regular” constraint functions: a function is regular if its value depends only on the multiset of input symbols, not on their order. This regularity captures many natural problems, including all multi‑state spin systems (e.g., Potts and Ising models), counting graph homomorphisms, weighted matchings, perfect matchings, the subgraph‑world formulation of the ferromagnetic Ising model, and any counting CSP or Holant problem with constant‑arity symmetric constraints.

The technical core consists of two complementary components. First, the authors design a fixed‑parameter tractable (FPT) exact algorithm for Holant instances whose underlying graph has bounded treewidth. By exploiting the symmetry of regular functions, the dynamic programming over a tree decomposition can be performed on a dramatically reduced state space: each bag needs only to record the multiplicities of domain values, leading to a runtime of O(f(k,q)·n) where k is the treewidth, q the domain size, and f depends solely on these parameters. Consequently, on graphs whose treewidth is bounded by a constant (or grows slowly), the exact Holant value can be computed in linear time.

Second, they lift this exact algorithm to planar and, more generally, apex‑minor‑free graphs by leveraging two structural facts. Apex‑minor‑free families are locally tree‑like: any large graph can be decomposed into pieces of small treewidth separated by a thin boundary. The second fact is the presence of strong spatial mixing (SSM) in the associated Gibbs measure. SSM guarantees that the influence of a boundary condition decays exponentially with distance, which allows one to approximate the effect of far‑away parts of the graph by fixing a local boundary with only a negligible error.

To prove SSM for the Holant setting, the authors extend the recursive coupling technique, previously used mainly for spin systems, to regular Holant constraints. They analyze the spectral radius of the transition matrix induced by the coupling and show that for ferromagnetic Potts models and the subgraph‑world problem the spectral radius is strictly less than one, establishing exponential decay of correlations. With SSM in hand, the algorithm proceeds recursively: it cuts the graph along a small separator, computes the exact Holant on each side using the treewidth‑bounded algorithm, and combines the results while controlling the accumulated error. By choosing the separator size and the approximation precision appropriately, the overall procedure runs in polynomial time in the input size and 1/ε, delivering a (1±ε)‑approximation – i.e., an FPTAS.

The paper then applies this general scheme to several concrete counting problems. For multi‑state ferromagnetic Potts models on planar graphs, it yields a deterministic FPTAS under the usual SSM condition (inverse temperature below the uniqueness threshold). For the subgraph‑world formulation of the ferromagnetic Ising model, the same result follows. Counting weighted matchings and perfect matchings, which correspond to specific Holant signatures, also become tractable on planar graphs. Moreover, counting graph homomorphisms with bounded-degree target graphs and many symmetric CSPs fall under the framework.

In summary, the authors demonstrate that for any Holant problem with regular symmetric constraints, strong spatial mixing implies the existence of a deterministic fully polynomial‑time approximation scheme on planar and apex‑minor‑free graphs. The work unifies a wide range of previously disparate counting problems, introduces a novel parameterized exact algorithm for bounded‑treewidth Holant instances, and extends recursive coupling to the Holant domain, thereby opening new avenues for deterministic approximation on sparse graph families.