[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82092-en":3,"doc-seo-82092-105":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},82092,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Binary Polynomial Optimization","Binary polynomial optimization (BPO) seeks the maximum of a multilinear polynomial defined over n binary variables, forming an NP-hard optimization class that includes unconstrained quadratic binary optimization (QUBO). The work generalizes prior tractable cases based on structural properties of hypergraphs, such as bounded tree-width, Berge-/β-/α-acyclicity, and related reach and rank constraints. A generalized variable elimination scheme is developed, yielding an explicit equivalent BPO polynomial after eliminating a chosen variable subset.","arXiv :2607 .08908v1 [math .CO] 9 Jul 2026  \nBinary Polynomial Optimization  \nEndre Boros∗  \nJuly 13, 2026  \nAbstract  \nIn a binary polynomial optimization problem (BPO, in short) we are maximizing a multilinear polynomial expression depending on n binary variables. This is a hard optimization class, containing many NP-hard problems, including unconstrained quadratic binary optimization. Several tractable special classes were considered in the literature, including problems with bounded tree-width (Crama, Hansen, Jaumard, 1990), Bergeacyclic problems (Buchheim, Crama, and Heck, 2019), β-acyclic problems (Del Pia and Di Gregorio, 2022, 2023), limited reach problems (Clausen, Crama, Lusby, Rodr´ıguez, and Ropke, 2024), and α-acyclic problems with bounded rank (Del Pia and Khajavirad, 2025) . We focus on a general variable elimination scheme for BPO, and develop the unique explicit multilinear polynomial form for the equivalent BPO problem obtained after the elimination of a given subset of the variables. The obtained closed form representation of such an equivalent BPO problem allows us to characterize new special classes for which this elimination method, when applied recursively, provides a computationally efficient solution. Our approach is elementary 1 , algebraic, and provides efficient solution to a wide problem class that properly generalizes all of the above mentioned tractable special cases.  \n1 Introduction  \nUnconstrained binary polynomial optimization (or in short BPO) aims at finding the maximum value of a polynomial expression depending on n binary variables. We use B = {0, 1} to denote the binary values, V = [n] = {1, 2 ,..., n} to denote the index set of the variables, and use xv ∈ B for v ∈ V for the binary variables in our problem. We denote by x = (xv | v ∈ V ) a vector of these binary variables, by BV the set of binary vectors, by R the set of real numbers, by Q the set of rational numbers, by Z the set of integers, and by Z + the set of nonnegative integers.  \n∗ MSIS Department and RUTCOR, Rutgers University, New Brunswick, New Jersey, USA. Email: [endre.boros@rutgers.edu](endre.boros@rutgers.edu).  \n1We do not use LP models and algorithms; this is a purely algebraic dynamic programming approach.  \nTo a given hypergraph H ⊆ 2V (that is a collection of subsets of V ) and rational vector c = (cH | H ∈ H) ∈ QH we associate a multilinear polynomial function  \nfH ,c (x) = X cH Y xv . (1)  \nH∈H v∈H  \nWith this notation, we can formulate the BPO problem as  \nmax fH c(x) . (2)  \nx∈BV ,  \nNote that for a binary value x ∈ B we have xk = x for all integers k ≥ 1, and thus any polynomial expression in these variables is equivalent to a multilinear one. Note also that H is not assumed to be a Sperner hypergraph, and in particular, it may contain a hyperedge of size 0 corresponding to a constant term, and hyperedges of size 1 corresponding to the linear terms of fH ,c , etc.  \nA mapping g : BV 7→ R is called a pseudo-Boolean function. It is wellknown that every pseudo-Boolean function has a unique multilinear polynomial representation with some hypergraph H ⊆ 2V and coefficients c ∈ RH , see [24, 25] . Problem BPO is a special case of pseudo Boolean optimization2 , and it contains unconstrained quadratic binary optimization (QUBO) as a special case. Thus, BPO is an NP-hard optimization class, since QUBO is known to include several well-known NP-complete combinatorial optimization problems, see e.g. [9, 29] for a more comprehensive list of related problems, equivalences, and applications.  \nThe structure of the hypergraph H plays an important role in the hardness/easiness of BPO. The so called basic-algorithm introduced by [23, 24] for pseudo-Boolean optimization was shown to be polynomial for BPO problems when H has bounded tree-width [15] . The standard linearization technique was shown to lead to an integral polytope, and hence to polynomial time solvability of BPO when H is Berge-acycic [12] . Unlike for graphs, acycli","cbCaiiewWsA5zD0D","https://ap.wps.com/l/cbCaiiewWsA5zD0D","pdf",420443,1,23,"English","en",105,"# Abstract\n# Introduction\n## Our Contributions","[{\"question\":\"What is the goal of a binary polynomial optimization (BPO) problem?\",\"answer\":\"BPO maximizes a multilinear polynomial expression defined over n binary variables. The objective depends on the chosen hypergraph structure and corresponding coefficients.\"},{\"question\":\"Why is BPO considered computationally hard?\",\"answer\":\"BPO is NP-hard because it generalizes unconstrained quadratic binary optimization (QUBO), which is known to contain NP-complete combinatorial optimization problems.\"},{\"question\":\"What is the main method developed in the work?\",\"answer\":\"The paper focuses on a generalized variable elimination scheme. It derives an explicit closed-form multilinear polynomial representation for the BPO instance obtained after eliminating a selected subset of variables.\"}]",1784178175,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"binary-polynomial-optimization","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/binary-polynomial-optimization/82092/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the goal of a binary polynomial optimization (BPO) problem?","Question",{"text":75,"@type":76},"BPO maximizes a multilinear polynomial expression defined over n binary variables. The objective depends on the chosen hypergraph structure and corresponding coefficients.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is BPO considered computationally hard?",{"text":80,"@type":76},"BPO is NP-hard because it generalizes unconstrained quadratic binary optimization (QUBO), which is known to contain NP-complete combinatorial optimization problems.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main method developed in the work?",{"text":84,"@type":76},"The paper focuses on a generalized variable elimination scheme. It derives an explicit closed-form multilinear polynomial representation for the BPO instance obtained after eliminating a selected subset of variables.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]