Near-Optimal Parallel Approximate Counting via Sampling

TL;DR

This paper introduces near-optimal parallel algorithms for approximate counting through sampling, reducing adaptivity and enabling efficient parallel computation for models like Ising and monomer-dimer.

Contribution

It presents non-adaptive and minimally adaptive algorithms that match the sample complexity of sequential methods, facilitating work-efficient parallel approximate counting.

Findings

Developed a non-adaptive approximate counting algorithm with $O(q \, \log^2 h / \varepsilon^2)$ samples.

Created an algorithm with $O(q \, \log h / \varepsilon^2)$ samples using only two rounds of adaptivity.

Applied these algorithms to achieve RNC counting for models like anti-ferromagnetic 2-spin, monomer-dimer, and ferromagnetic Ising.

Abstract

The computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is achieved via simulated annealing, where the counting problem is formulated in terms of estimating the ratio $Q={Z(\beta_{\max})}/{Z(\beta_{\min})}$ between partition functions $Z(\beta)=\sum_{x\in \Omega} \exp(\beta H(x))$ of Gibbs distributions $\mu_\beta$ over $\Omega$ with Hamiltonian $H$, given access to a sampling oracle that produces samples from $\mu_\beta$ for $\beta \in [\beta_{\min}, \beta_{\max}]$. The best bound achieved by known annealing algorithms with relative error $\varepsilon$ is $O(q \log h / \varepsilon^2)$, where $q, h$ are parameters which respectively bound $\ln Q$ and $H$. However, all known algorithms attaining this near-optimal complexity are inherently sequential, or *adaptive*: the queried parameters $\beta$ depend on previous samples. We develop a simple non-adaptive algorithm for approximate counting using $O(q \log^2 h / \varepsilon^2)$ samples, as well as an algorithm that achieves $O(q \log h / \varepsilon^2)$ samples with just two rounds of adaptivity, matching the best sample complexity of sequential algorithms. These algorithms naturally give rise to work-efficient parallel (RNC) counting algorithms. We discuss applications to RNC counting algorithms for several classic models, including the anti-ferromagnetic 2-spin, monomer-dimer and ferromagnetic Ising models.