[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82921-en":3,"doc-seo-82921-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82921,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Fast Counting and Sampling for Ferromagnetic Two-Spin Systems","The paper develops two equivalent formulations for ferromagnetic two-spin systems: a weighted subgraph model and a random-cluster-type model. These connections enable an efficient sampling algorithm and a randomized counting algorithm that approximates the partition function in specific parameter regimes. The proposed estimation method achieves near-quadratic runtime on bounded-degree graphs and polynomial time on general graphs. The work improves prior bounds by Fengarving parameter regimes where earlier methods lacked efficient sampling.","arXiv :2607 .05248v 1 [ cs .DS] 6 Jul 2026  \nFAST COUNTING AND SAMPLING FOR FERROMAGNETIC TWO-SPIN SYSTEMS  \nWEIMING FENG, HENG GUO, AND YICHUN YANG  \nAbstract. We introduce two new models equivalent to ferromagnetic two-spin systems: a weighted subgraph model and a random cluster type model. Using these new connections, we obtain an efficient sampling algorithm and a new randomised algorithm that efficiently approximates the partition function of ferromagnetic two-spin systems in certain parameter regimes. No efficient sampling algorithms are known before in this regime, and our new estimation algorithm runs in near-quadratic time for bounded degree graphs and in polynomial time for general graphs, improving upon the previous algorithm of Guo, Liu, and Lu (2020) .  \n1. Introduction  \nSpin systems model nearest neighbour interactions. They originate from statistical physics, and are now widely studied in other areas including probability theory, machine learning, and theoretical computer science, sometimes under different names such as Markov random fields. The most important computation task in their study is to estimate the so-called partition function, which is a central quantity linking to many of the system’s macroscopic properties.  \nA particularly important case is when the number of spin is 2 . These two-state spin systems (or 2-spin systems for short) show drastically different behaviours depending on whether the interaction is repulsive (anti-ferromagnetic) or attractive (ferromagnetic) . For anti-ferromagnetic systems, great progress has been made regarding partition function estimation – it is NP-hard beyond the tree-uniqueness threshold [Sly10, SS14, GV16], and has an efficient algorithm otherwise [Wei06, SST14, LLY13, CLV23, CCYZ25] . The complexity landscape for the anti-ferromagnetic case is completely mapped out.  \nOn the other hand, ferromagnetic systems have demonstrated a more involved complexity landscape. Here the most well-known system is the Ising model, and the pioneer work of Jerrum and Sinclair [JS93] showed that efficient algorithms for the ferromagnetic Ising model exist regardless of the tree-uniqueness threshold. Their approach is direct Markov chain Monte Carlo (MCMC), and this approach was taken further by Goldberg, Jerrum, and Paterson [GJP03] for general ferromagnetic 2-spin systems under certain conditions. Later, Guo and Lu [GL18] applied Weitz’s correlation decay method [Wei06] to push the algorithmic threshold even further. In certain regimes, their algorithm is almost optimal, up to a threshold beyond which the problem becomes \\#BIS-hard [LLZ14] . Here \\#BIS stands for the problem of counting independent sets in bipartite graphs, which is conjectured to be intractable to approximate [DGGJ04] . However, Guo and Lu’s algorithm does not extend to the case where both spins are attracting. This is not a barrier to algorithms though, as subsequently Guo, Liu, and Lu [GLL20] successfully adapted Barvinok’s method [Bar16, PR17] to this setting. These works [JS93, GJP03, GL18, GLL20] constitute the current algorithmic frontier for ferromagnetic 2-spin systems, but there is still a large gap between known algorithmic and hardness thresholds.  \nA main drawback of the correlation decay method and Barvinok’s method is that the running time of the resulting algorithm is often a polynomial with a large exponent. In particular, the exponent for the algorithm in [GLL20] scales logarithmically in the maximum degree, and it runs within polynomial-time only for bounded degree graphs. In this paper, we address this issue by giving a new algorithm that runs in  \n(Weiming Feng) School of Computing and Data Science, The University of Hong Kong, Hong Kong, China  \n(Heng Guo) School of Informatics, University of Edinburgh, Informatics Forum, Edinburgh, EH8 9AB, United Kingdom  \n(Yichun Yang) School of Computer Science, Beijing Institute of Technology, Beijing, China  \nnear-quadratic time for bounded degree graphs","cbCaikE4Q6oHzGxE","https://ap.wps.com/l/cbCaikE4Q6oHzGxE","pdf",880221,3,1,36,"English","en",105,"# Introduction\n## Sampling and counting for ferromagnetic two-spin systems","[{\"question\":\"What two new equivalent models does the paper introduce for ferromagnetic two-spin systems?\",\"answer\":\"It introduces a weighted subgraph model and a random cluster type model, each equivalent to the original ferromagnetic two-spin system formulation.\"},{\"question\":\"What performance guarantees are given for approximating the partition function?\",\"answer\":\"The randomized estimation algorithm runs in near-quadratic time for bounded-degree graphs and in polynomial time for general graphs, under certain parameter regimes.\"},{\"question\":\"Why does the paper claim earlier algorithms could not provide efficient sampling in the relevant setting?\",\"answer\":\"It notes that due to lack of self-reducibility, no efficient sampling algorithm was known before, and it resolves this by using an intermediate random-cluster-type model to convert samples between formulations.\"}]",1784183967,91,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"fast-counting-and-sampling-for-ferromagnetic-two-spin-systems","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/fast-counting-and-sampling-for-ferromagnetic-two-spin-systems/82921/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-23","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What two new equivalent models does the paper introduce for ferromagnetic two-spin systems?","Question",{"text":75,"@type":76},"It introduces a weighted subgraph model and a random cluster type model, each equivalent to the original ferromagnetic two-spin system formulation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What performance guarantees are given for approximating the partition function?",{"text":80,"@type":76},"The randomized estimation algorithm runs in near-quadratic time for bounded-degree graphs and in polynomial time for general graphs, under certain parameter regimes.",{"name":82,"@type":73,"acceptedAnswer":83},"Why does the paper claim earlier algorithms could not provide efficient sampling in the relevant setting?",{"text":84,"@type":76},"It notes that due to lack of self-reducibility, no efficient sampling algorithm was known before, and it resolves this by using an intermediate random-cluster-type model to convert samples between 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