Deterministic counting from coupling independence
Xiaoyu Chen, Weiming Feng, Heng Guo, Xinyuan Zhang, Zongrui Zou
TL;DR
This work delivers a deterministic FPTAS for counting partition functions of spin systems on bounded-degree graphs under coupling independence and a marginal lower bound, matching the best randomized results in several colourings regimes. The key innovation is a recursive marginal estimator that combines a constant-size LP-based partial coupling with exponential total-influence decay to reduce complex global dependencies to manageable local computations. By linking contractive Markov-chain couplings to coupling independence, the authors obtain CI results for colourings in high-degree regimes and under the Dobrushin–Shlosman condition, unifying several known algorithmic thresholds under a deterministic framework. The approach yields FPTASes for q-colourings on high girth graphs and for general bounded-degree graphs in multiple parameter regimes, offering a practical deterministic alternative to existing randomized approximations.
Abstract
We show that spin systems with bounded degrees and coupling independence admit fully polynomial time approximation schemes (FPTAS). We design a new recursive deterministic counting algorithm to achieve this. As applications, we give the first FPTASes for $q$-colourings on graphs of bounded maximum degree $Δ\ge 3$, when $q\ge (11/6-\varepsilon_0)Δ$ for some small $\varepsilon_0\approx 10^{-5}$, or when $Δ\ge 125$ and $q\ge 1.809Δ$, and on graphs with sufficiently large (but constant) girth, when $q\geqΔ+3$. These bounds match the current best randomised approximate counting algorithms by Chen, Delcourt, Moitra, Perarnau, and Postle (2019), Carlson and Vigoda (2024), and Chen, Liu, Mani, and Moitra (2023), respectively.