Problem
Research questions and friction points this paper is trying to address.
monomer-dimer model
near-linear time
sampling
parallel sampling
approximate counting
Sampling Matchings in Near-linear Time
🤖 AI Summary
该研究解决了单体-二聚体模型中的匹配采样问题，通过近线性时间的Glauber动力学实现了高效采样、并行处理和快速计数。
📝 Abstract
For every fixed activity $λ>0$, we establish three results for the monomer--dimer model on an $n$-vertex simple graph $G$ with $m\ge1$ edges and maximum degree $Δ$.
1. Near-linear mixing and sampling. Single-edge Glauber dynamics has mixing time $O_λ(m[\log^2 n+\log(1/\varepsilon)])$, giving a near-linear-time approximate sampler.
2. Work-efficient parallel sampling. We simulate the same Glauber dynamics in parallel using $\tilde{O}_λ(m+n)$ work and $\tilde{O}_λ(\min\{Δ,m^{1/3},\sqrt n\})$ depth with high probability.
3. Fast approximate counting. We estimate the partition function within relative error $\varepsilon$ in $\tilde{O}_λ(n^2/\varepsilon^2)$ work. For dense graphs with $m=Θ(n^2)$, this is near-linear in the input size.
For the mixing theorem, we establish a general log--Sobolev criterion based on field-dynamics spectral stability, with only logarithmic dependence on the inverse occupied-marginal lower bound. Parallelism uses a matching-specific analysis of occupation-interval dependencies. Counting uses monomer-preconditioned Jerrum--Sinclair dynamics, whose parameters are learned efficiently by Glauber dynamics.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Near-linear time sampling
Work-efficient parallel sampling
Fast approximate counting
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