[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85555-en":3,"doc-seo-85555-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85555,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Near-Optimal Parallel Approximate Counting via Sampling","Computational equivalence between approximate counting and sampling is leveraged to build faster algorithms for estimating partition ratios. Prior reductions based on simulated annealing yield near-optimal sequential performance, but they require inherently adaptive or sequential queried parameters. This work proposes a simple non-adaptive approximate counting method using O(qε−2 log2 h) samples, and an almost-matching variant with two additional adaptive rounds achieving O(qε−2 log h) samples. The resulting approach yields work-efficient parallel RNC counting algorithms, applied to classic spin and matching models.","arXiv :2604 .0 1263v2 [ cs .DS] 12 Jul 2026  \nNear-Optimal Parallel Approximate Counting via Sampling David G. Harris∗ Vladimir Kolmogorov† Hongyang Liu‡ Yitong Yin‡ Yiyao Zhang‡  \nAbstract  \nThe computational equivalence between approximate counting and sampling is well established for polynomial-time algorithms. The most efficient general reduction from counting to sampling is based on simulated annealing. In this approach, the counting problem is formulated as estimating the ratio Q = Z (βmax ) /Z (β min) between partition functions Z(β) = ∑x∈Ωexp(βH(x)) of Gibbs distributions µ β over Ω with Hamiltonian H, given access to a sampling oracle for µ β at any β ∈ [β min, βmax ] . The sample complexity (measured by the number of oracle calls) is typically expressed in terms of q and h, which respectively bound ln Q and H. The best upper bound achieved by known annealing algorithms with relative error ε is O(qε−2 log h) . However, all known algorithms attaining this near-optimal complexity are inherently sequential, or adaptive: the queried parameters β depend on previous samples.  \nWe develop a simple non-adaptive algorithm for approximate counting using O(qε−2 log2 h) samples, as well as an algorithm that achieves O(qε−2 log h) samples with just two additional adaptive rounds, matching the best sample complexity of sequential algorithms. These algorithms naturally yield work-efficient parallel (RNC) counting algorithms. We discuss applications to several classic models, including the anti-ferromagnetic 2-spin, monomer-dimer and ferromagnetic Ising models.  \n1 Introduction  \nA fundamental theme in randomized computation is the intrinsic connection between sampling combinatorial objects and counting those objects. This interplay culminated in the Monte Carlo method and has become a central paradigm in algorithm design. In theoretical computer science, a classic result of Jerrum, Valiant and Vazirani [JVV86] established that, for all self-reducible problems, approximate counting and sampling are computationally equivalent up to polynomial time.  \nA wide range of counting problems can be formulated as computing the partition function of a Gibbs distribution, for which significantly more efficient approximation algorithms have been developed by leveraging samples from the corresponding Gibbs distributions. Classic examples include volume estimation [DFK91], approximation of the permanent [JSV04], and approximately counting combinatorial objects such as matchings, independent sets, and spin configurations [JS93; LV97], or constraint-satisfying solutions [FGYZ21] .  \n∗ Department of Computer Science, University of Maryland, College Park, [USA.](USA. davidgharris29@gmail.com)[ davidgharris29@gmail.com](USA. davidgharris29@gmail.com)[ ](USA. davidgharris29@gmail.com)†Institute of Science and Technology Austria, Klosterneuburg, [Austria.](Austria. vnk@ist.ac.at)[ vnk@ist.ac.at](Austria. vnk@ist.ac.at)  \n‡School of Computer Science, State Key Laboratory for Novel Software Technology, New Cornerstone Science Laboratory, Nanjing University, Nanjing, China. {liuhongyang, [zhangyiyao}@smail.nju.edu.cn](zhangyiyao}@smail.nju.edu.cn), [yinyt@nju.edu.cn](yinyt@nju.edu.cn)  \nFormally, given a real-valued function H(·) over a finite set Ω, and a weight function F : Ω → R≥0, the Gibbs distribution is the family of distributions µΩβ over Ω, parameterized by β over an interval [β min, βmax ], of the form  \nµΩβ(ω) = eβHZ(ω()Fβ)(ω) .  \nThe normalization factor Z(β) = ∑ω ∈ΩF(ω)eβH(ω) is called the partition function. These distributions appear in a number of sampling algorithms and are also common in physics, where the parameter −β corresponds to the inverse temperature, and H(ω) is called the Hamiltonian of the system. In the unweighted setting, we have F(ω) = 1 for all ω . Given boundary values [β min, βmax ], a key parameter is the partition ratio Q = Z (βmax ) /Z (β min) .  \nSince our algorithms only use the sampled value of the Hamiltonian, it is c","cbCaieflEHeCvRCW","https://ap.wps.com/l/cbCaieflEHeCvRCW","pdf",488483,1,23,"English","en",105,"# Introduction\n## Estimating the Partition Ratio","[{\"question\":\"What is the core problem addressed in the document?\",\"answer\":\"The document focuses on estimating the partition ratio Q = Z(βmax)/Z(βmin) for Gibbs distributions, which enables approximate counting of combinatorial objects.\"},{\"question\":\"How do the proposed algorithms improve over prior annealing-based reductions?\",\"answer\":\"They avoid fully sequential adaptive queries by introducing a non-adaptive sampling approach and a variant with only two additional adaptive rounds, while matching near-optimal sample complexity.\"},{\"question\":\"What types of models are discussed as applications?\",\"answer\":\"The document discusses applications to anti-ferromagnetic 2-spin, monomer-dimer, and ferromagnetic Ising models, framed through partition functions and Gibbs 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is the core problem addressed in the document?","Question",{"text":75,"@type":76},"The document focuses on estimating the partition ratio Q = Z(βmax)/Z(βmin) for Gibbs distributions, which enables approximate counting of combinatorial objects.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the proposed algorithms improve over prior annealing-based reductions?",{"text":80,"@type":76},"They avoid fully sequential adaptive queries by introducing a non-adaptive sampling approach and a variant with only two additional adaptive rounds, while matching near-optimal sample complexity.",{"name":82,"@type":73,"acceptedAnswer":83},"What types of models are discussed as applications?",{"text":84,"@type":76},"The document discusses applications to anti-ferromagnetic 2-spin, monomer-dimer, and ferromagnetic Ising models, framed through partition functions and Gibbs 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