Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Subquadratic Counting via Perfect Marginal Sampling

Xiaoyu Chen, Zongchen Chen, Kuikui Liu, Xinyuan Zhang

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.

Subquadratic Counting via Perfect Marginal Sampling

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 -time algorithms, where 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 -weighted independent sets in graphs of maximum degree , we obtain a -time approximate counting algorithm, for some constant , when the fugacity , improving over the previous regime of 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 counting and 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 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.

Paper Structure

This paper contains 40 sections, 35 theorems, 91 equations, 10 algorithms.

Table of Contents

  1. Introduction
  2. Main Results
  3. Technical Overview
  4. Faster Variance Decay for an Unbiased Estimator
  5. Aggregate Perfect Marginal Sampler
  6. Prior Work
  7. On Subquadratic Counting
  8. On Approximate Counting in Spin Systems
  9. On Perfect Marginal Samplers
  10. Preliminaries
  11. Notations
  12. Distributions on Product Spaces
  13. Multinomial Distributions
  14. Graphs
  15. Models
  16. ...and 25 more sections

Key Result

Theorem 1.1

Let $\Delta \ge 3$ be an integer and $\lambda < \frac{1}{\Delta-1}$ be a real number. Then, there exists an algorithm that, for any $n$-vertex graph $G$ of maximum degree $\Delta$ and any $\varepsilon>0$, approximates the partition function of the hardcore model on $G$ with fugacity $\lambda$ with $

Theorems & Definitions (73)

  • Theorem 1.1: Informal version of \ref{['corollary:hardcore']}; see also \ref{['thm:2-spin-reduction']}
  • Theorem 1.2: Informal
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6: Informal version of \ref{['thm:batch-automaton']}
  • Remark 1.7
  • Definition 2.1
  • Definition 2.2: perfect marginal sampling
  • Remark 2.3
  • ...and 63 more