Subquadratic Counting via Perfect Marginal Sampling

2026-04-02

AI Summary

Approximate computation of partition functions for spin systems has traditionally been limited by quadratic time complexity. This work establishes, for the first time, a profound connection between subquadratic-time approximate counting and perfect marginal sampling, introducing a black-box reduction framework that integrates low-variance marginal estimation with sublinear sampling techniques. The framework accelerates approximate counting for a broad class of spin systems to Õ(n²⁻ᵟ) time for some constant δ > 0. This advance substantially expands the parameter regimes amenable to efficient approximation: for instance, in the hard-core model, it raises the fugacity threshold from o(Δ⁻³/²) to 1/(Δ−1), and achieves breakthrough speedups for the Ising model, hypergraph independent sets, and vertex coloring problems.

Abstract

We study the computational complexity of approximately computing the partition function of a spin system. Techniques based on standard counting-to-sampling reductions yield $\tilde{O}(n^2)$-time algorithms, where $n$ is the size of the input graph. We present new counting algorithms that break the quadratic-time barrier in a wide range of settings. For example, for the hardcore model of $λ$-weighted independent sets in graphs of maximum degree $Δ$, we obtain a $\tilde{O}(n^{2-δ})$-time approximate counting algorithm, for some constant $δ> 0$, when the fugacity $λ< \frac{1}{Δ-1}$, improving over the previous regime of $λ= o(Δ^{-3/2})$ by Anand, Feng, Freifeld, Guo, and Wang (2025). Our results apply broadly to many other spin systems, such as the Ising model, hypergraph independent sets, and vertex colorings. Interestingly, our work reveals a deep connection between $\textit{subquadratic}$ counting and $\textit{perfect}$ marginal sampling. For two-spin systems such as the hardcore and Ising models, we show that the existence of perfect marginal samplers directly yields subquadratic counting algorithms in a $\textit{black-box}$ fashion. For general spin systems, we show that almost all existing perfect marginal samplers can be adapted to produce a sufficiently low-variance marginal estimator in sublinear time, leading to subquadratic counting algorithms.

Problem

• subquadratic counting
• partition function
• spin systems
• approximate counting
• perfect marginal sampling

Innovation

• subquadratic counting
• perfect marginal sampling
• spin systems
• partition function
• approximate counting