[Paper Review] Approximate Counting via Correlation Decay on Planar Graphs

Yitong Yin, Chihao Zhang|arXiv (Cornell University)|Jul 16, 2012

Markov Chains and Monte Carlo Methods4 citations

TL;DR

This paper establishes that correlation decay (strong spatial mixing) implies a deterministic fully polynomial-time approximation scheme (FPTAS) for a broad class of Holant problems on planar and apex-minor-free graphs. By developing a fixed-parameter tractable algorithm for exact computation on bounded-treewidth graphs and extending recursive coupling techniques, the authors prove FPTAS for problems like the ferromagnetic Potts model and subgraphs world under specific parameter conditions, unifying approximation results across diverse counting problems.

ABSTRACT

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.

Motivation & Objective

• To establish a general connection between correlation decay and efficient approximation for Holant problems on restricted graph families.
• To extend the recursive coupling technique to Holant problems, enabling strong spatial mixing proofs for models like the ferromagnetic Potts model.
• To develop a fixed-parameter tractable algorithm for exact Holant computation on bounded-treewidth graphs with regular constraint functions.
• To prove that strong spatial mixing implies FPTAS for Holant problems on apex-minor-free graphs, including planar graphs.
• To unify approximation results across diverse counting problems—such as spin systems, graph homomorphisms, and weighted matchings—under a single framework.

Proposed method

• Introduces a notion of regularity for constraint functions in Holant problems, covering symmetric functions of constant arity and diverse counting problems.
• Develops a fixed-parameter tractable algorithm to compute exact Holant values on graphs of bounded treewidth, using dynamic programming over tree decompositions.
• Leverages the locally tree-like structure of apex-minor-free graphs to extend exact algorithms to approximate solutions via correlation decay.
• Applies recursive coupling techniques to prove strong spatial mixing for the ferromagnetic Potts model and subgraphs world problem.
• Uses the self-reduction equivalence between marginal probabilities and partition functions to relate correlation decay to approximability.
• Derives explicit parameter thresholds (in terms of degree Δ, inverse temperature β, and domain size q) under which FPTAS exists.

Experimental results

Research questions

• RQ1 Under what conditions does correlation decay imply an FPTAS for Holant problems on planar and apex-minor-free graphs?
• RQ2 Can the recursive coupling method be generalized to Holant problems beyond spin systems?
• RQ3 What is the relationship between strong spatial mixing and the tractability of approximate counting in Holant problems with regular constraint functions?
• RQ4 For which parameter regimes of the ferromagnetic Potts model and subgraphs world problem does an FPTAS exist on apex-minor-free graphs?
• RQ5 How can fixed-parameter tractable algorithms on bounded-treewidth graphs be extended to yield approximation schemes on more general graph families?

Key findings

• Strong spatial mixing in the Gibbs measure of a Holant problem implies the existence of an FPTAS on apex-minor-free graphs when the constraint functions are regular.
• An FPTAS exists for the subgraphs world problem with parameters 0 < μ, λ < 1 if the maximum degree Δ satisfies Δ < (1 + λμ²)² / (1 − μ²).
• An FPTAS exists for the ferromagnetic Ising model with inverse temperature β and external field B if Δ < (e²ᵇ⁺⁴ᴮ + e²ᵇ + 2e²ᴮ)² / [e²ᴮ(e²ᵇ + 1)²(e²ᴮ + 1)²].
• An FPTAS exists for the q-state ferromagnetic Potts model if β < ln((q−2)/(Δ−1)) / (Δ+1), under the condition that q−2 > (λ−1)(Δ−1)λ^Δ.
• The recursive coupling technique is successfully extended to Holant problems, enabling the proof of strong spatial mixing for the ferromagnetic Potts model.
• The framework unifies approximation algorithms for diverse problems including spin systems, graph homomorphisms, and counting CSPs under a single theoretical umbrella.