[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82725-en":3,"doc-seo-82725-105":30,"detail-sidebar-cat-0-en-105":92},{"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},82725,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","An FPT Algorithm for Diverse Minimum s–t Cuts","The study investigates selecting k minimum s–t edge cuts in a directed weighted graph such that every pair differs by at least d edges, measured via the symmetric difference. For d in {1,2}, the task matches counting minimum s–t cuts, known to be #P-complete, and is already NP-complete for k=3. The paper proves the problem is fixed-parameter tractable when parameterized by k+d. The FPT approach relies on new structural properties and a flow-augmentation technique adapted from prior work.","arXiv :2607 .03266v 1 [ cs .DS] 3 Jul 2026  \nAn FPT Algorithm for Diverse Minimum s–t Cuts  \nKrishnan Dehaleesan∗ P˚al Grøn˚as Drange† Fedor V. Fomin‡  \nPetr A. Golovach§ Laure Morelle ¶  \nAbstract  \nWe study the problem of finding a family of diverse minimum edge s–t cuts in a directed weighted graph G. Given integers k and d, the task is to decide whether G contains k minimum s–t cuts C1 ,..., Ck such that for any i, j ∈ [k], the number of edges in the symmetric difference Ci △Cj is at least d.  \nFor d ∈ {1, 2}, the problem corresponds to counting minimum s–t cuts in G, which is \\#P-complete [Provan and Ball, SICOMP 1983] . The problem is also known to be NP-complete already for k = 3 [de Berg, L´opez Mart´ınez, Spieksma, ISAAC 2024] . Our main result shows that the problem is fixed-parameter tractable (FPT) when parameterized by the combined parameter k + d.  \nThe main ingredients of our FPT algorithm build on novel structural properties of diverse minimums–t cuts and a non-trivial application of the flow-augmentation technique of Kim, Kratsch, Pilipczuk, and Wahlstr¨om [JACM 2025] .  \n1 Introduction  \nThe minimum s–t cut problem is one of the most fundamental and central problems in graph algorithms. Given a directed graph with designated vertices s and t, the task is to separate s from t by removing a set of edges of minimum total capacity. This problem is deeply connected to the theory of network flows, as formalized by the max-flow min-cut theorem, and to notions of graph connectivity.  \nIt is well known that the minimum s–t cut problem is solvable in polynomial time via the connection to the Maximum Flow problem, and a long line of research has culminated in an almost-linear time O (m1+o(1)) [5] . However, several natural variants quickly become computationally more challenging. In particular, counting the number of minimum s–t cuts is \\#P-hard [21] . On the other hand, the maximum number of disjoint minimum s–t cuts could be easily found in polynomial time [25] . (One proceeds iteratively by computing the leftmost minimum cut and contracting its edges.)  \nIn applications, one often wants not just a single minimum s–t cut, but several optimal alternatives from which a user can choose. A naive approach may return cuts that are almost identical, differing in only a few edges. It is therefore natural to ask for a family of minimum s–t cuts that is diverse, meaning that every pair differs in at least d > 0 edges. This perspective fits into the broader study of diversity in combinatorial optimization [4, 11 , 12 , 15 , 8] . In this context, de Berg et al. [7] introduced the notion of diverse minimum s–t cuts. The problem of identifying diverse minimum s–t cuts occupies an “intermediate” position between counting all minimum cuts (\\#P-complete) and computing pairwise disjoint minimum cuts (in P) .  \nFor a family of minimum s–t cuts C1 , C2 ,..., Ck ⊆ E (G) in a graph G, de Berg et al. define the following natural notions of their diversity:  \n∗ University of Bergen, Norway. Email: [krishnan.dehaleesan@uib.no](krishnan.dehaleesan@uib.no).  \n†University of Bergen, Norway. Email: [Pal.Drange@uib.no](Pal.Drange@uib.no).  \n‡University of Bergen, Norway. Email: [fedor.fomin@uib.no](fedor.fomin@uib.no. Research)[. Research](fedor.fomin@uib.no. Research) supported by the Research Council of Norway under BWCA project (grant no. 314528) and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (NewPC grant agreement No. 101199930)  \n§ University of Bergen, Norway. Email: [petr.golovach@uib.no](petr.golovach@uib.no. Research)[. Research](petr.golovach@uib.no. Research) supported by the Research Council of Norway under the BWCA (grant no. 314528) and Extreme-Algorithms (grant no 355137) projects.  \n¶ University of Bergen, Norway. Email: [Laure.Morelle@uib.no](Laure.Morelle@uib.no. Research)[. Research](Laure.Morelle@uib.no. Research) supported by the Research Council of Norway u","cbCaiksus3tiJ4Gu","https://ap.wps.com/l/cbCaiksus3tiJ4Gu","pdf",566159,6,1,14,"English","en",105,"# Introduction\n## Overview of Our Results","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses deciding whether a directed weighted graph contains k minimum s–t cuts whose pairwise symmetric differences contain at least d edges.\"},{\"question\":\"Why is the problem computationally hard in general?\",\"answer\":\"For d in {1,2} it corresponds to counting minimum s–t cuts, which is #P-complete, and it is NP-complete already when k=3.\"},{\"question\":\"What is the main algorithmic contribution?\",\"answer\":\"The paper shows the diverse minimum s–t cuts problem is fixed-parameter tractable when parameterized by the combined parameter k+d, using new structural properties and a flow-augmentation technique.\"}]",1784182516,35,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"an-fpt-algorithm-for-diverse-minimum-st-cuts","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/an-fpt-algorithm-for-diverse-minimum-st-cuts/82725/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does the paper address?","Question",{"text":76,"@type":77},"It addresses deciding whether a directed weighted graph contains k minimum s–t cuts whose pairwise symmetric differences contain at least d edges.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why is the problem computationally hard in general?",{"text":81,"@type":77},"For d in {1,2} it corresponds to counting minimum s–t cuts, which is #P-complete, and it is NP-complete already when k=3.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the main algorithmic contribution?",{"text":85,"@type":77},"The paper shows the diverse minimum s–t cuts problem is fixed-parameter tractable when parameterized by the combined parameter k+d, using new structural properties and a flow-augmentation technique.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"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":107,"slug":138},19,"General","general"]