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Work-Efficient Parallel Counting via Sampling

Hongyang Liu, Yitong Yin, Yiyao Zhang

TL;DR

This work tackles the fundamental problem of counting by reducing it to sampling for Gibbs partition functions. It develops a parallel simulated-annealing approach with a non-adaptive cooling schedule that achieves near-optimal total work while providing logarithmic-depth parallelism, through Product and Paired Product estimators and a BoundaryFind subroutine. The main result is a parallel algorithm with total cost O($q$ ε^{-2} log^2 h) and depth O(log $q$ + log log h + log 1/ε), applicable in a single oracle round, and it yields work-efficient RNC counting for key graphical models—hardcore and Ising—in their uniqueness regimes. These contributions enable scalable, provably efficient counting via sampling for fundamental statistical physics models, with concrete parallel bounds for practical n-vertex graphs.

Abstract

A canonical approach to approximating the partition function of a Gibbs distribution via sampling is simulated annealing. This method has led to efficient reductions from counting to sampling, including: $\bullet$ classic non-adaptive (parallel) algorithms with sub-optimal cost (Dyer-Frieze-Kannan '89; Bezáková-Štefankovič-Vazirani-Vigoda '08); $\bullet$ adaptive (sequential) algorithms with near-optimal cost (Štefankovič-Vempala-Vigoda '09; Huber '15; Kolmogorov '18; Harris-Kolmogorov '24). We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.

Work-Efficient Parallel Counting via Sampling

TL;DR

This work tackles the fundamental problem of counting by reducing it to sampling for Gibbs partition functions. It develops a parallel simulated-annealing approach with a non-adaptive cooling schedule that achieves near-optimal total work while providing logarithmic-depth parallelism, through Product and Paired Product estimators and a BoundaryFind subroutine. The main result is a parallel algorithm with total cost O( ε^{-2} log^2 h) and depth O(log + log log h + log 1/ε), applicable in a single oracle round, and it yields work-efficient RNC counting for key graphical models—hardcore and Ising—in their uniqueness regimes. These contributions enable scalable, provably efficient counting via sampling for fundamental statistical physics models, with concrete parallel bounds for practical n-vertex graphs.

Abstract

A canonical approach to approximating the partition function of a Gibbs distribution via sampling is simulated annealing. This method has led to efficient reductions from counting to sampling, including: classic non-adaptive (parallel) algorithms with sub-optimal cost (Dyer-Frieze-Kannan '89; Bezáková-Štefankovič-Vazirani-Vigoda '08); adaptive (sequential) algorithms with near-optimal cost (Štefankovič-Vempala-Vigoda '09; Huber '15; Kolmogorov '18; Harris-Kolmogorov '24). We present an algorithm that achieves both near-optimal total work and efficient parallelism, providing a reduction from counting to sampling with logarithmic depth and near-optimal work. As consequences, we obtain work-efficient parallel counting algorithms for several important models, including the hardcore and Ising models within the uniqueness regime.
Paper Structure (19 sections, 11 theorems, 19 equations, 1 table, 4 algorithms)

This paper contains 19 sections, 11 theorems, 19 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Partition function of Gibbs distribution
  3. Graphic models
  4. Hardcore model.
  5. Ising model.
  6. Simulated annealing
  7. Our results
  8. Applications.
  9. Preliminaries
  10. Chebyshev's inequality
  11. Monotonicity and convexity of partition function
  12. Parallel Annealing Algorithm
  13. The cooling schedule.
  14. The classic Product Estimator (PE).
  15. The Paired Product Estimator (PPE).
  16. ...and 4 more sections

Key Result

Theorem 1.7

Fix any Hamiltonian function $H: \Omega \rightarrow \left\{0, 1, \ldots, h\right\}$ and let $q=\ln|\Omega|$. For any given values of $0\le \beta_{\min}<\beta_{\max}$, assuming a sampling oracle $\mathcal{O}$ for the interval $[\beta_{\min},\beta_{\max}]$, there exists a parallel algorithm that, give with total cost of $O(q \varepsilon^{-2} \log^2 h)$ and depth of $O(\log q + \log \log h + \log \va

Theorems & Definitions (20)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7: Main result
  • Remark 1.8: Near optimality in total work
  • Corollary 1.9: Hardcore models
  • Corollary 1.10: Ising models
  • ...and 10 more