[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86509-en":3,"doc-seo-86509-105":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},86509,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Constant-factor Approximation of MinCostCSP with a Conservative Majority Polymorphism","Minimum Cost Constraint Satisfaction Problem (MinCostCSP) seeks an assignment that minimizes the sum of rational variable-value costs under constraints from a fixed relational structure A. While exact classification of MinCostCSP(A) is known and constant-factor approximability fails when A lacks a conservative near-unanimity polymorphism, this work studies structures admitting a conservative majority (3-near-unanimity) polymorphism. It proves an unavoidable dichotomy criterion, moving toward a broader classification of constant-factor approximation complexity.","Constant-factor approximation of MinCostCSP with a conservative majority polymorphism  \narXiv :2607 . 10667v 1 [ cs .CC] 12 Jul 2026  \nMarcin Kozik Jagiellonian University [marcin. kozik@uj. edu. pl](marcin. kozik@uj. edu. pl)  \nStanislav Živný University of Oxford [standa. zivny@cs. ox. ac. uk](standa. zivny@cs. ox. ac. uk)  \nAbstract  \nFor a relational structure A, the Minimum Cost Constraint Satisfaction Problem is the following problem denoted by MinCostCSP (A): Given an instance of CSP (A) with rational costs on variable-value pairs, find a solution to the instance minimizing the sum of the chosen costs. For the exact minimization, a classification of MinCostCSP (A) in terms of A was established by Takhanov [STACS’10] .  \nWe focus on constant-factor approximations of MinCostCSP (A) . DeHaan, Huang, and Lee recently showed that if A fails to admit a conservative near-unanimity polymorphism then MinCostCSP (A) is not constant-factor approximable [APPROX’25] . We provide a first step towards a classification, by proving a dichotomy for structures A admitting a conservative majority (also known as 3-near-unanimity) polymorphism. Our dichotomy criterion is not in terms of an algebraic condition on A but we show that this is unavoidable. We include a simple argument proving that no such condition exists.  \n1 Introduction  \nSatisfiability of formulas with three literals per clause (3-SAT), Maximum Cut (MaxCut), and Minimum Vertex Cover (MinVC) are all examples of Boolean Constraint Satisfaction Problems (CSPs) [17] . Given a set of variables and a set of constraints, one seeks an assignment of 0sand 1s to the variables with the goal of satisfying all the constraints (3-SAT), maximizing the number of satisfied constraints (MaxCut), and minimizing the number of variables assigned the value 1 while simultaneously satisfying all the constraints (MinVC) . 1  \nThis paper is concerned with CSPs parameterized by the set of allowed constraint relations, known as non-uniform CSPs [46], fixed-template CSPs [27], or CSPs with a fixed constraint language [39] . Formally, one fixes a (relational) structure A = (A; R1 , . . . , Rp), where A is a finite domain of values for the variables and each Ri is a relation on A. We denote by CSP (A) the set of CSP instances whose constraints only use relations from A. For example, 3-SAT is captured by a CSP with eight (in fact four suffice) relations corresponding to the eight types of 3-clauses, MaxCut is captured by a CSP with one relation, namely the binary disequality relation {(0, 1) ,(1 , 0)}, and MinVC is captured by a CSP with the relation {(0, 1) ,(1 , 0) ,(1 , 1)} . The three examples deal with different computational problems. 3-SAT is an example of a decision Boolean CSP. All decision Boolean CSPs were classified as solvable in polynomial time or NP-complete by Schaefer back in 1978 [58] . MaxCut is an example of an optimization Boolean CSP. All optimization Boolean CSPs were classified with respect to exact solvability  \n1We call a CSP Boolean if the domain of each variable is of size two.  \nby Creignou [16] and with respect to approximability, both for minimization and maximization, by Khanna, Sudan, Trevisan, and Williamson [43] (see also the monograph by Creignou, Khanna, and Sudan [17]) . Finally, MinVC is one of the simplest examples of a Boolean CSP that mixes decision and optimization. In other words, it is an optimization problem with strict constraints. Such CSPs are called MinOnes in [43, 17] and TMIN by Khanna and Motwani in [42] . All Boolean MinOnes, as well as MaxOnes, were classified with respect to approximability in [43] .  \nWhile by now we have a very good understanding of Boolean CSPs, the situation is significantly more complicated for CSPs over larger, non-Boolean domains, also known as alphabets. Over the years, various fragments of non-Boolean CSPs have been studied. Two prominent examples that were important in the development of the field are graph CSPs and conservat","cbCainK9nbeySGUT","https://ap.wps.com/l/cbCainK9nbeySGUT","pdf",634361,3,1,36,"English","en",105,"# Abstract\n# Introduction\n## CSPs and related optimization/minimization problems\n## Non-Boolean CSPs and known classifications\n## Approximation differences: MaxCSP vs MinCSP","[{\"question\":\"What problem does MinCostCSP(A) define?\",\"answer\":\"MinCostCSP(A) takes a CSP instance with rational costs on variable-value pairs and asks for a satisfying assignment minimizing the total sum of chosen costs.\"},{\"question\":\"What prior result limits constant-factor approximations for MinCostCSP(A)?\",\"answer\":\"If A does not admit a conservative near-unanimity polymorphism, then MinCostCSP(A) is not constant-factor approximable.\"},{\"question\":\"What does this paper contribute for structures with a conservative majority polymorphism?\",\"answer\":\"It proves a dichotomy criterion for structures A that admit a conservative majority (3-near-unanimity) polymorphism, and argues that no purely algebraic condition of A can capture the full criterion.\"}]",1784212281,91,{"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},"constant-factor-approximation-of-mincostcsp-with-a-conservative-majority-polymorphism","",{"@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/constant-factor-approximation-of-mincostcsp-with-a-conservative-majority-polymorphism/86509/",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","2026-07-27","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 problem does MinCostCSP(A) define?","Question",{"text":75,"@type":76},"MinCostCSP(A) takes a CSP instance with rational costs on variable-value pairs and asks for a satisfying assignment minimizing the total sum of chosen costs.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What prior result limits constant-factor approximations for MinCostCSP(A)?",{"text":80,"@type":76},"If A does not admit a conservative near-unanimity polymorphism, then MinCostCSP(A) is not constant-factor approximable.",{"name":82,"@type":73,"acceptedAnswer":83},"What does this paper contribute for structures with a conservative majority polymorphism?",{"text":84,"@type":76},"It proves a dichotomy criterion for structures A that admit a conservative majority (3-near-unanimity) polymorphism, and argues that no purely algebraic condition of A can capture the full 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