[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86345-en":3,"doc-seo-86345-105":30,"detail-sidebar-cat-0-en-105":83},{"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},86345,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Partially Ordered Sets Corresponding to the Partition Problem","The partition problem is addressed through its optimization version by constructing two partially ordered sets (posets) and an order-theoretic framework. One poset is order-isomorphic to a known structure tied to subset-sum solutions, while the second poset is a crucial subposet. Dominance relations between subsets are characterized uniformly across instances. Properties such as size, height, and width are analyzed, with matching width Θ(2^n/n^{3/2}) revealing exponential hardness. Initial optimal candidates correspond to elements of the second poset, enabling further reductions and polynomially solvable cases.","arXiv :2405 .05544v2 [ cs .DM] 11 Jul 2026  \nPARTIALLY ORDERED SETS CORRESPONDING TO THE  \nPARTITION PROBLEM∗  \nSUSUMU KUBO†  \nAbstract. The partition problem is a well-known NP-complete problem. We focus on its optimization version. We propose two partially ordered sets (posets) corresponding to the partition problem and develop an order-theoretic framework for solving it. The first poset is order-isomorphic to a well-known poset whose structure is related to solutions of the subset sum problem, while the second is a subposet of the first and plays a crucial role in this paper. The partial order characterizes the dominance relations between subsets that hold uniformly across all instances. We first show several properties of the two posets, such as size, height, and width (the size of the largest antichain, i.e., the largest set of pairwise incomparable elements) . The two posets have the same width, which is Θ(2n /n3/2) for n congruent to 0 or 3 modulo 4; this exponential width indicates the hardness of the partition problem. We then prove that the initial candidate solutions are the elements of the second poset, whose size is 2n −2 􀀀 ⌊nn/2⌋ 􀀁 . Since a partition corresponds to two elements of the poset, the number of initial candidate partitions is half of that, i.e., 2n−1 −􀀀 ⌊nn/2⌋ 􀀁 . We prove that the candidate solutions can be further reduced based on the partial order, and we establish a necessary and sufficient condition, phrased in terms of the second poset, for a subset to attain the optimal value. Building on this optimality criterion, we finally derive several polynomially solvable cases from the structure of the second poset. Our approach offers a useful tool for structural analysis of the partition problem.  \nKey words. partial order, partition problem, subset sum problem, combinatorial optimization, Sperner property  \nMSC codes. 06A06, 06A07, 90C27, 68Q25  \n1. Introduction. The partition problem is formulated as follows: given positive integers c 1 , . . . , cn , the objective is to decide whether there exists a subset S of {1,..., n} =: [n] such that ∆(S) = 0, where  \n(1.1) ∆(S) :=X ci − X ci.  \ni∈S i∈[n]\\S  \nIn this paper, we focus on its optimization version: to find a subset S of [n] that minimizes the absolute value |∆(S)| . Without loss of generality, we index the integers in nonincreasing order (c1 ≥ c2 ≥ · · · ≥ cn) . Although the ci’s are originally positive, in what follows we only require cn ≥ 0; allowing zero entries loses no generality and is convenient because some proofs use instances with zero entries.  \nThe partition problem is one of Karp’s 21 NP-complete problems [14] and is also one of Garey and Johnson’s six basic NP-complete problems [7] . Therefore, the optimization version is known to be NP-hard. It has important applications in task scheduling [4, 31] . In fact, it is equivalent to makespan minimization on two identical parallel machines.  \nThe problem has been investigated from various perspectives. Several optimal algorithms give exact solutions in time exponential in n [7, 11, 16, 26] . On the pseudopolynomial side, the classical dynamic programming approach has been improved  \n∗ An extended abstract version of this paper appeared in Proceedings of the 25th International Symposium on Fundamentals of Computation Theory, 2025, [https://doi.org/10.1007/](https://doi.org/10.1007/)[ ](https://doi.org/10.1007/)[978-3-032-04700-7](978-3-032-04700-7 23)[ ](978-3-032-04700-7 23)[23](978-3-032-04700-7 23).  \nFunding: This work was partially supported by MEXT Leading Initiative for Excellent Young Researchers Grant Number JPMXS0320200347 and JSPS KAKENHI Grant Number JP26K17039 .  \n†Faculty of Informatics, Showa Women’s University, Tokyo, Japan ([s-kubo@swu.ac.jp](s-kubo@swu.ac.jp)) .  \n2 S. KUBO  \nto near-linear time in the total input sum [3, 15] . Moreover, a variety of heuristic and metaheuristic algorithms have been developed [1, 6, 12, 13, 25] . An “easy-hard”phase transition was observed","cbCaihLJ0QicSlBt","https://ap.wps.com/l/cbCaihLJ0QicSlBt","pdf",525061,4,1,28,"English","en",105,"# Introduction\n## Main Results","[{\"question\":\"Which performance complexity indicator is used to express the difficulty of the partition problem?\",\"answer\":\"The width of the posets, given by Θ(2^n/n^{3/2}) for specific congruence classes of n, is shown to reflect exponential hardness. The exponential number of incomparable elements (largest antichain) drives this conclusion.\"}]",1784210699,71,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"partially-ordered-sets-corresponding-to-the-partition-problem","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/partially-ordered-sets-corresponding-to-the-partition-problem/86345/",{"url":52,"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","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Which performance complexity indicator is used to express the difficulty of the partition problem?","Question",{"text":75,"@type":76},"The width of the posets, given by Θ(2^n/n^{3/2}) for specific congruence classes of n, is shown to reflect exponential hardness. The exponential number of incomparable elements (largest antichain) drives this conclusion.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]