[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85591-en":3,"doc-seo-85591-105":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},85591,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Branch width of represented matroids in matrix multiplication time","For an n-element matroid M given by a n×m matrix representation over a finite field F and an integer k, an algorithm is presented with running time O(k m,F(n^2)) for finding either a branch-decomposition of M of width at most k or a proof that the branch-width exceeds k, assuming k\u003Cα with α=2.3714 the matrix multiplication exponent. The hidden factors depend computably on k and F. Previous methods have cubic-time bottlenecks, but a standard-form input enables a faster bound by shifting overhead to standard-form computation. Consequences include faster rank-width algorithms for directed graphs and pathwidth algorithms for fixed-field matroid representations, plus an approximation approach on infinite fields.","Branch-width of represented matroids in matrix  \nmultiplication time  \nMujin Choi∗2,1, Tuukka Korhonen†3, and Sang-il Oum∗1,2  \n1 Discrete Mathematics Group, Institute for Basic Science (IBS),  \nDaejeon, South Korea  \n2 Department of Mathematical Sciences, KAIST, Daejeon, South Korea  \n3University of Copenhagen, Copenhagen, Denmark Email [addresses:](addresses: mujinchoi@kaist.ac.kr)[ mujinchoi@kaist.ac.kr](addresses: mujinchoi@kaist.ac.kr), [tuko@di.ku.dk](tuko@di.ku.dk), [sangil@ibs.re.kr](sangil@ibs.re.kr)  \narXiv :2605 . 14428v2 [ cs .DS] 13 Jul 2026  \nJuly 14, 2026  \nAbstract  \nFor an 􀀽-element matroid 􀀢 given by an 􀀽 × 􀀽 matrix representation over a finite field Fand an integer 􀀺, we present an algorithm with running time 􀀤 􀀺,F(􀀽2 )+􀀤(􀀽􀁬 ) that either finds a branch-decomposition of 􀀢 of width at most 􀀺, or confirms that the branch-width of􀀢 is more than 􀀺, where 􀁬 \u003C 2. 3714 is the matrix multiplication exponent, and the 􀀤 􀀺,F(·)-notation hides factors that depend on 􀀺 and F in a computable manner. All previous algorithms, including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)], have cubic-time bottlenecks. Moreover, if the input matrix representation is given in standard form, our algorithm runs in 􀀤 􀀺,F(􀀽2 ) time, since 􀀤(􀀽􀁬 ) time is only needed for finding a standard form of the input matrix. When 􀀢 is given by an 􀀼 × 􀀽 matrix, the overhead for finding a standard form is 􀀤(􀀼􀀽 min( 􀀼, 􀀽) 􀁬 −2) .  \nAs corollaries, we obtain faster algorithms for rank-width of directed graphs and pathwidth of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is in standard form and contains a bounded number of distinct values of entries.  \nTo suggest that our algorithm is optimal, we observe that for every field F, deciding whether the branch-width of a matroid represented over F is 0 is as hard as deciding whether a square matrix over F is singular. Under the assumption that singularity testing requires Ω (􀀽􀁬 )-time, this implies that the overhead of 􀀤(􀀽􀁬 ) is unavoidable. We also show strengthenings ofthis observation to rule out some approximations under this assumption.  \n1 Introduction  \nA matroid is a pair 􀀢 = (􀀚, I) of a finite set 􀀚 and a collection I of subsets of 􀀚 called independent  \nsets such that ∅ ∈ I, all subsets of independent sets are independent, and all maximal independent ∗ Supported by the Institute for Basic Science (IBS-R029-C1)  \n†Supported by the European Union under Marie Skłodowska-Curie Actions (MSCA), project no. 101206430, and by the VILLUM Foundation, Grant Number 54451, Basic Algorithms Research Copenhagen (BARC) .  \n1  \nCo-funded by  \nthe European Union  \nsubsets of a set 􀀭 have the same size, denoted by 􀁁 (􀀭 ), the rank function of 􀀢 . For a matrix 􀀖 over a field F whose columns are indexed by a set 􀀚, if I is the collection of all subsets of 􀀚 whose corresponding column vectors of 􀀖 are linearly independent, then 􀀢(􀀖) = (􀀚, I) is a matroid. Ifa matroid 􀀢 admits a matrix 􀀖 over a field F such that 􀀢 = 􀀢 (􀀖), then it is representable over F. We say a matroid is represented over F ifit is given with its matrix representation over F.  \nBranch-width was introduced by Robertson and Seymour [24] for graphs, hypergraphs, matroids, and more generally, for connectivity functions. Here we will describe its definition formatroids. Let 􀀢 be a matroid on a finite set 􀀚 with the rank function 􀁁 . The connectivity function of a matroid 􀀢 is defined as 􀁟 􀀢(􀀭 ) := 􀁁 (􀀭 ) +􀁁(􀀚 \\􀀭 ) −􀁁(􀀚) . It is well known that 􀁟 􀀢 is symmetric, meaning that 􀁟 􀀢(􀀭 ) = 􀁟 􀀢(􀀚 \\ 􀀭 ) for all 􀀭 ⊆ 􀀚, and submodular, meaning that  \n􀁟 􀀢 (􀀭 ) + 􀁟 􀀢 (􀀮 ) ≥ 􀁟 􀀢 (􀀭 ∩ 􀀮 ) + 􀁟 􀀢 (􀀭 ∪ 􀀮 )  \nfor all 􀀭, 􀀮 ⊆ 􀀚 . Roughly speaking, the branch-width of a matroid 􀀢 is the minimum 􀀺 such that its ground set can be recursively partitioned into two parts until singletons re","cbCaiivx2MAxnfs8","https://ap.wps.com/l/cbCaiivx2MAxnfs8","pdf",445604,3,1,30,"English","en",105,"# Abstract\n# Introduction\n## Matroid representation and rank\n## Branch-width and connectivity function\n## Complexity and existing frameworks\n## Relation to tree-width algorithms","[{\"question\":\"What is the main problem addressed by the algorithm?\",\"answer\":\"Given a matroid represented over a finite field by a matrix and a target integer k, the algorithm either constructs a branch-decomposition of width at most k or certifies that the branch-width is greater than k.\"},{\"question\":\"What running time does the proposed algorithm achieve?\",\"answer\":\"The running time is expressed in terms of matrix multiplication performance: it is O(k m,F(n^2))+O(n^k) in the paper’s notation, where the factors depending on k and F are computable.\"},{\"question\":\"Why is standard form of the input matrix important?\",\"answer\":\"When the input matrix representation is in standard form, the algorithm avoids the additional cost needed to find a standard form, making the overall running time faster.\"}]",1784204788,76,{"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},"branch-width-of-represented-matroids-in-matrix-multiplication-time","",{"@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/branch-width-of-represented-matroids-in-matrix-multiplication-time/85591/",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","2026-07-25","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 main problem addressed by the algorithm?","Question",{"text":75,"@type":76},"Given a matroid represented over a finite field by a matrix and a target integer k, the algorithm either constructs a branch-decomposition of width at most k or certifies that the branch-width is greater than k.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What running time does the proposed algorithm achieve?",{"text":80,"@type":76},"The running time is expressed in terms of matrix multiplication performance: it is O(k m,F(n^2))+O(n^k) in the paper’s notation, where the factors depending on k and F are computable.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is standard form of the input matrix important?",{"text":84,"@type":76},"When the input matrix representation is in standard form, the algorithm avoids the additional cost needed to find a standard form, making the overall running time 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