Subquadratic Counting via Perfect Marginal Sampling
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arXiv
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| Format: | Preprint |
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2026
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| author | Chen, Xiaoyu Chen, Zongchen Liu, Kuikui Zhang, Xinyuan |
| author_facet | Chen, Xiaoyu Chen, Zongchen Liu, Kuikui Zhang, Xinyuan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02235 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Subquadratic Counting via Perfect Marginal Sampling Chen, Xiaoyu Chen, Zongchen Liu, Kuikui Zhang, Xinyuan Data Structures and Algorithms Probability 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. |
| title | Subquadratic Counting via Perfect Marginal Sampling |
| topic | Data Structures and Algorithms Probability |
| url | https://arxiv.org/abs/2604.02235 |