[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83196-en":3,"doc-seo-83196-105":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},83196,1374391974468,"Eden","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Ranking and Rank Aggregation with Matroid Prefix Constraints","Ranking and rank aggregation are studied using Kendall tau distance while imposing matroid or flag matroid constraints on prefixes of the output ranking. In the matroid setting, the top-k prefix must form a matroid base; in the flag matroid setting, multiple prescribed prefixes must form bases of a sequence of matroids connected by quotient relations. A polynomial-time greedy algorithm finds the closest feasible ranking under flag matroid prefix constraints. Optimality is proven via a Bruhat order argument, and approximation frameworks extend. Rank aggregation with matroid constraints is NP-hard for any fixed m ≥ 2 inputs.","arXiv :2607 .07 153v 1 [ cs .DM] 8 Jul 2026  \nRanking and Rank Aggregation with Matroid Prefix Constraints  \nSeiei Ando∗ Yu Yokoi†  \nAbstract  \nWe study ranking and rank aggregation under the Kendall tau distance, subject tomatroid or flag matroid constraints on prefixes of the output ranking. In the matroid case, the top-k prefix is required to form a base of a matroid; in the flag matroid case, several prescribed prefixes are required to form bases of a sequence of matroids linked by quotient relations. This framework contains the previously studied notions of k-fairness and blockfairness as special cases, and also captures more general hierarchical and assignment-type lower-and upper-quota constraints.  \nWe provide a polynomial-time algorithm for finding, given a single input ranking, a closest feasible ranking under flag matroid prefix constraints. The algorithm is a natural greedy procedure, and its optimality is proved via a Bruhat order argument on the symmetric group. As a consequence, existing approximation frameworks for fair rank aggregation carry over to the matroidal setting. We also prove that rank aggregation with matroid constraints is NP-hard for every fixed number m ≥ 2 of input rankings, even under partition matroid constraints.  \n1 Introduction  \nRanking a set of alternatives and aggregating multiple rankings into a single consensus ranking are fundamental tasks in social choice, information retrieval, meta-search, recommendation, and many other applications [19 , 36] . Throughout the paper, a ranking of a finite set E is a total order on E , which we identify with a bijection π : [n] → E where [n] = {1,..., n} , n = |E|, and π (i) is the element placed at position i.  \nTo compare rankings, we use the Kendall tau distance (a.k.a. bubble-sort distance) . For two rankings π and σ, their Kendall tau distance, denoted by dKT (π,σ), is the number of unordered pairs of elements on which π and σ disagree. This is one of the most standard distances in rank aggregation and voting theory. In the unconstrained setting, minimizing the sum of Kendall tau distances to the input rankings yields the classical Kemeny rule [29 , 39 , 40] . The Kemeny aggregation rule has strong axiomatic justification, but its computational side is difficult: computing an optimal Kemeny ranking is NP-hard [3 , 25] . On the positive side, constant-factor approximations and even a PTAS are known [1 , 30] .  \nIn this paper, we study constrained versions of ranking and rank aggregation problems. One  \nnatural source of constraints is fairness. Recent work on fair ranking has imposed group-fair representation requirements on top positions of a ranking. For instance, Celis–Straszak–Vishnoi [14] studied top-position fairness under metrics other than Kendall tau. In the Kendall tau setting,∗ Department of Mathematical and Computing Science, School of Computing, Institute of Science Tokyo,  \nTokyo 152-8552, Japan. Email: [ando.s.373d@m.isct.ac.jp](ando.s.373d@m.isct.ac.jp)  \n†Department of Mathematical and Computing Science, School of Computing, Institute of Science Tokyo,  \nTokyo 152-8552, Japan. Email: [yokoi@comp.isct.ac.jp](yokoi@comp.isct.ac.jp)  \nthe closest fair ranking (CFR) problem and the fair rank aggregation (FRA) problem have been studied under proportionate fairness notions [16 , 37] . In the model of Chakraborty–Das–Khan– Subramanian [16], the candidate set is partitioned into groups, and k-fairness requires the top-k prefix to satisfy lower and upper bounds on the number of candidates from each group. The stronger notion of block-fairness imposes analogous constraints on multiple prescribed prefixes. Under these notions, CFR admits exact polynomial-time algorithms, whereas FRA has mainly been studied via approximation algorithms [15 , 16 , 37] .  \nOur starting point is the observation that the fairness constraints studied in CFR and FRA have a matroidal structure. The k-fairness constraint can be expressed as the requirement that the ","cbCaivzyl1JzUZBr","https://ap.wps.com/l/cbCaivzyl1JzUZBr","pdf",820573,2,1,34,"English","en",105,"# Introduction\n## Background on ranking and Kendall tau\n## Fair ranking and matroid structure\n# Contributions\n## Closest ranking under flag matroid constraints\n## From matroid-CFR to weighted base problems","[{\"question\":\"What distance measure is used for comparing rankings in this work?\",\"answer\":\"The paper uses Kendall tau distance (bubble-sort distance), counting unordered pairs on which two rankings disagree.\"},{\"question\":\"How do matroid and flag matroid prefix constraints differ?\",\"answer\":\"For matroid constraints, the top-k prefix must form a base of a matroid. For flag matroid constraints, several prescribed prefixes must each form bases of matroids in a flag linked by quotient relations.\"},{\"question\":\"Is there an efficient algorithm for the closest feasible ranking under flag matroid constraints?\",\"answer\":\"Yes. The paper provides a polynomial-time greedy algorithm to find, from a single input ranking, a closest feasible ranking under flag matroid prefix constraints, with optimality proved via a Bruhat order argument.\"}]",1784185893,86,{"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},"ranking-and-rank-aggregation-with-matroid-prefix-constraints","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/ranking-and-rank-aggregation-with-matroid-prefix-constraints/83196/",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 distance measure is used for comparing rankings in this work?","Question",{"text":75,"@type":76},"The paper uses Kendall tau distance (bubble-sort distance), counting unordered pairs on which two rankings disagree.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do matroid and flag matroid prefix constraints differ?",{"text":80,"@type":76},"For matroid constraints, the top-k prefix must form a base of a matroid. For flag matroid constraints, several prescribed prefixes must each form bases of matroids in a flag linked by quotient relations.",{"name":82,"@type":73,"acceptedAnswer":83},"Is there an efficient algorithm for the closest feasible ranking under flag matroid constraints?",{"text":84,"@type":76},"Yes. 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