[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83238-en":3,"doc-seo-83238-105":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},83238,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Revisiting Maximum k-Biplex Search Through k-Bounded-Degree Deletion","Biplex relaxes the biclique model and serves as a cohesive subgraph notion for bipartite graphs. The maximum k-biplex search problem seeks a k-biplex with the maximum number of edges but is NP-hard, and exact methods become inefficient on large bipartite graphs with k≥3. The work establishes a structural duality: maximizing a k-biplex corresponds to finding a minimal k-bounded-degree deletion in the complement graph. A deletion-based algorithm is proposed, with proven worst-case complexity O*(αk) where α1=1.725, α2=1.856, and α3=1.928, plus pruning and heuristics that greatly reduce search space.","Revisiting Maximum 􀀺-Biplex Search Through 􀀺-Bounded-Degree  \nDeletion  \nDonghang Cui†, Rong-Hua Li†, Qiangqiang Dai†, Guoren Wang†  \n†Beijing Institute of Technology, China  \n{[cuidonghang@bit.edu.cn](cuidonghang@bit.edu.cn), [lironghuabit@126.com](lironghuabit@126.com), [qiangd66@gmail.com](qiangd66@gmail.com),[wanggrbit@126.com}](wanggrbit@126.com})  \narXiv :2607 .074 19v 1 [ cs .DS] 8 Jul 2026  \nAbstract  \nBiplex, as a relaxation of the biclique model, has emerged as an important cohesive subgraph model for bipartite graph analysis. The maximum 􀀺-biplex search problem aims to identify the 􀀺-biplex with maximum number of edges and has been widely applied in various real-world applications, including community detection, online recommendation, and fraud detection. However, the problem is NP-hard, and existing exact algorithms remain inefficient on large-scale bipartite graphs with large values of 􀀺 (e.g., 􀀺 ≥ 3) . In this paper, we revisit the maximum 􀀺-biplex search problem from a complementary perspective. We reveal a novel structural duality: finding a maximum 􀀺-biplex in a bipartite graph is equivalent to finding a minimal 􀀺-bounded-degree deletion in its complement graph. Based on this observation, we propose a novel deletionbased algorithm for the maximum 􀀺-biplex search problem. We theoretically prove that the proposed algorithm achieves a worstcase time complexity of 􀀤∗(􀁗􀀽􀀺) , where 􀁗􀀺 \u003C 2. Specifically, 􀁗 1 = 1.725, 􀁗2 = 1. 856, and 􀁗3 = 1.928. To further enhance practical efficiency, we develop several effective upper-bounding techniques and a heuristic strategy for obtaining high-quality initial solutions, which substantially reduce the search space. Extensive experiments on eight real-world bipartite graphs demonstrate the efficiency of our approach, which achieves up to four orders of magnitude speedups over state-of-the-art algorithms.  \n1 Introduction  \nBipartite graphs are fundamental data structures for modeling interactions between two distinct groups of entities, such as user-item interactions in e-commerce networks [21, 35], author-publication relationships in academic networks [20], and gene-protein associations in computational biology networks [9, 31, 34]. Identifying cohesive subgraphs is a core task in bipartite graph mining, since such subgraphs often reveal meaningful groups with strong crossside associations. For example, in computational biology networks, cohesive subgraphs can capture groups of genes and proteins with strong expression associations, thereby helping identify functional modules or disease-related biological pathways [34] .  \nAs a classical cohesive subgraph model in bipartite graphs, the biclique has been widely studied and applied in many scenarios [4, 17, 18] . A biclique is a complete bipartite subgraph, where every vertex on one side is connected to all vertices on the other side. The maximum biclique search problem aims to find a biclique containing the maximum number of edges, which has attracted considerable attention in recent years [2, 8, 10, 12, 23, 26] . However, real-world network data are inevitably affected by observation noise, measurement errors, and missing data [22, 24] . Due to the strict complete-connectivity requirement, the biclique is unable to  \nrepresent cohesive communities that contain a small number of missing edges, which limits its practical application.  \nTo address this limitation, the 􀀺-biplex model was introduced as a relaxation of the biclique and has attracted increasing attention in recent years [7, 8, 25, 27, 35–37] . Formally, in a 􀀺-biplex of a bipartite graph, each vertex has at most 􀀺 non-neighbors. This relaxation allows limited missing edges while preserving its cohesion, and has been applied to many real-world tasks such as online recommendation [13, 28], community detection [8, 14], and fraud detection [8, 35] . In this paper, we study the problem of finding a 􀀺-biplex with the maximum number of edges, which is referred to as the maxim","cbCaivAZgz7VVA6h","https://ap.wps.com/l/cbCaivAZgz7VVA6h","pdf",975468,3,1,13,"English","en",105,"# Introduction\n## Biplex and k-biplex as cohesive subgraph models\n## Complexity of the maximum k-biplex search problem\n## Exact algorithms and their limitations\n## Relation to k-biplex enumeration\n## Contributions and duality idea","[{\"question\":\"What problem does the maximum k-biplex search aim to solve?\",\"answer\":\"It aims to find a k-biplex in a bipartite graph that has the maximum number of edges. The paper notes this problem is NP-hard for any positive integer k.\"},{\"question\":\"What structural duality does the paper reveal?\",\"answer\":\"Finding a maximum k-biplex in a bipartite graph is equivalent to finding a minimal k-bounded-degree deletion in its complement graph. This enables the proposed deletion-based approach.\"},{\"question\":\"How does the proposed algorithm improve performance compared with existing exact methods?\",\"answer\":\"The paper proposes a deletion-based algorithm with a proven worst-case time complexity and adds upper-bounding techniques and a heuristic for high-quality initial solutions. Experiments on eight real-world graphs show speedups up to four orders of magnitude.\"}]",1784186152,33,{"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},"revisiting-maximum-k-biplex-search-through-k-bounded-degree-deletion","",{"@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/revisiting-maximum-k-biplex-search-through-k-bounded-degree-deletion/83238/",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","2026-07-24","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 the maximum k-biplex search aim to solve?","Question",{"text":75,"@type":76},"It aims to find a k-biplex in a bipartite graph that has the maximum number of edges. The paper notes this problem is NP-hard for any positive integer k.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What structural duality does the paper reveal?",{"text":80,"@type":76},"Finding a maximum k-biplex in a bipartite graph is equivalent to finding a minimal k-bounded-degree deletion in its complement graph. This enables the proposed deletion-based approach.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed algorithm improve performance compared with existing exact methods?",{"text":84,"@type":76},"The paper proposes a deletion-based algorithm with a proven worst-case time complexity and adds upper-bounding techniques and a heuristic for high-quality initial solutions. Experiments on eight real-world graphs show speedups up to four orders of magnitude.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]