[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85609-en":3,"doc-seo-85609-105":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},85609,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","O(n)+f(k): Truly Linear FPT","Parameterized complexity focuses on practical computation by isolating combinatorial blow-up into a secondary parameter k. With today’s Big Data, even quadratic dependence on the input size n can be prohibitive, changing what counts as a practically efficient algorithm. The work defines Truly Linear FPT (TLFPT) as O(n)+f(k) and proves via diagonalization that it forms a strict subset of Linear FPT (LFPT) defined as O(n)·f(k). It also develops techniques using treedepth and BFS-width and shows TLFPT inhabitants across numerous graph and combinatorial problems.","arXiv :2606 .02492v 1 [ cs .CC] 1 Jun 2026  \nO (n) + f(k): Truly Linear FPT  \nBenjamin Merlin Bumpus \\#   \nInstituto de Matemática e Estatística, Universidade de São Paulo. Rua do Matão, 1010— 05508–090, São Paulo, SP, Brasil.  \nRod Downey \\#   \nVictoria University of Wellington, School of Mathematics and Statistics, PO Box 600 Wellington, New Zealand  \nTala Eagling-Vose \\#   \nDepartment of Computer Science, Durham University, Durham, UK Jessica Enright \\#   \nSchool of Computing Science, University of Glasgow Michael R. Fellows \\#   \nInstitute of Informatics, University of Bergen (Emeritus), Bergen, Norway David C. Kutner \\#   \nDepartment of Computer Science, Durham University, Durham, UK Laura Larios-Jones \\#   \nSchool of Computing Science, University of Glasgow, Glasgow, UK Barnaby Martin \\#   \nDepartment of Computer Science, Durham University, Durham, UK Frances Rosamond \\#   \nInstitute of Informatics, University of Bergen (Emerita), Bergen, Norway Ella Yates \\#   \nSchool of Computing Science, University of Glasgow, Glasgow, UK  \n~~ Abstract ~~  \nParameterized complexity has always been concerned with practical computing: by confining combinatorial explosion to a secondary parameter k, one can uncover why and how many NP-hard problems are effectively tackled in practice. Today, however, the scale of data has changed: scientists study Big Data, which is so large that even quadratic dependence in the total input size n isunaffordable. Therefore, what constitutes a practical algorithm has also changed. Classically, parameterized complexity is blind to the difference between defining fixed parameter tractability multiplicatively (i.e. f (k) · nc ) or additively (i.e. f (k) + nc ) . But what if the constant c is one and we require true linearity, is this distinction still inconsequential? Here, we define and explore Truly Linear FPT (TLFPT)– that is O (n) + f(k)– and show that it is a strict subset of Linear FPT (LFPT)– that is O (n) · f(k)– via diagonalization.  \nPopulating TLFPT requires careful consideration of linear-time algorithmics and data structures. We meet many inhabitants of TLFPT: SAT, Vertex Cover, Min-Max Matching, (n − k) -Coloring, Diverse Pair of Matchings, k-Path, and H-Coloring. Our parameterizations are equally varied. Beyond classical parameters like solution size, we leverage two parameters, treedepth and BFS-width, which are particularly well-suited to the TLFPT regime. We do so by developing techniques based on depth-and breadth-first search.  \nFor parameterized complexity to be of service to the scientific community, we need to contend with Big Data. For sufficiently large inputs, FPT beyond linear may not suffice. Thus, there is a practical and theoretical need for more ambitious goals. TLFPT is a first step forward.  \n2012 ACM Subject Classification Theory of computation → Fixed parameter tractability; Theory of computation → Complexity classes; Theory of computation → Parameterized complexity and exact algorithms; Mathematics of computing → Graph algorithms; Theory of computation → Dynamic programming; Theory of computation → Streaming, sublinear and near linear time algorithms  \n© Benjamin Bumpus, Rod Downey, Tala Eagling-Vose, Jessica Enright, Michael R. Fellows, David C. Kutner, Laura Larios-Jones, Barnaby Martin, Frances Rosamond, and Ella Yates;  \nlicensed under Creative Commons License CC-BY 4.0  \nLeibniz International Proceedings in Informatics  \nSchloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany  \nXX:2 O (n) + f(k): Truly Linear FPT  \nKeywords and phrases Truly Linear FPT, fixed-parameter tractability, parameterized complexity, graph algorithms, kernelization, dynamic programming.  \nDigital Object Identifier 10.4230/LIPIcs...  \nFunding Jessica Enright: Partly supported by EPSRC grants EP/T004878/1 and EP/V032305/1 David C. Kutner: Partly supported by EPSRC grant EP/T004878/1, Multilayer Algorithmics to Leverage Graph Structure (MultilayerALGS)  .  \nBarnaby Martin:","cbCaiezqEXKLPQtt","https://ap.wps.com/l/cbCaiezqEXKLPQtt","pdf",1607669,3,1,42,"English","en",105,"# Introduction\n## Truly Linear FPT vs Linear FPT\n## Parameters and key techniques\n## Big Data motivation","[{\"question\":\"What is Truly Linear FPT (TLFPT) and how does it differ from Linear FPT (LFPT)?\",\"answer\":\"TLFPT requires running time O(n)+f(k), while LFPT allows O(n)·f(k). The document proves TLFPT is a strict subset of LFPT using diagonalization.\"},{\"question\":\"Why is the motivation for TLFPT tied to Big Data?\",\"answer\":\"With massive inputs, even dependence like n^2 becomes unaffordable, so practicality depends on near-linear dependence on n. The document argues that classical multiplicative vs additive distinctions are no longer sufficient at this scale.\"},{\"question\":\"Which graph problems and parameters are presented as fitting the TLFPT regime?\",\"answer\":\"The work lists SAT, Vertex Cover, Min-Max Matching, (n−k)-Coloring, Diverse Pair of Matchings, k-Path, and H-Coloring. It uses parameters including treedepth and BFS-width and develops methods based on depth and breadth-first search.\"}]",1784204896,106,{"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},"onfk-truly-linear-fpt","",{"@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/onfk-truly-linear-fpt/85609/",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 Truly Linear FPT (TLFPT) and how does it differ from Linear FPT (LFPT)?","Question",{"text":75,"@type":76},"TLFPT requires running time O(n)+f(k), while LFPT allows O(n)·f(k). The document proves TLFPT is a strict subset of LFPT using diagonalization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why is the motivation for TLFPT tied to Big Data?",{"text":80,"@type":76},"With massive inputs, even dependence like n^2 becomes unaffordable, so practicality depends on near-linear dependence on n. The document argues that classical multiplicative vs additive distinctions are no longer sufficient at this scale.",{"name":82,"@type":73,"acceptedAnswer":83},"Which graph problems and parameters are presented as fitting the TLFPT regime?",{"text":84,"@type":76},"The work lists SAT, Vertex Cover, Min-Max Matching, (n−k)-Coloring, Diverse Pair of Matchings, k-Path, and H-Coloring. It uses parameters including treedepth and BFS-width and develops methods based on depth and breadth-first search.","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"]