[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86544-en":3,"doc-seo-86544-105":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},86544,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Courcelle’s Theorem in Truly Linear FPT","The paper studies Truly Linear FPT (TLFPT), a parameterized complexity class defined by algorithms running in O(n)+f(k) time. It addresses open questions left by prior work by presenting a general method to design TLFPT algorithms parameterized by treewidth. The main contribution is a TLFPT algorithm for Courcelle’s theorem: given a graph G, integer k, and a CMSO2 formula φ, the method runs in O(n+m)+f(k,φ) and either certifies that tw(G)>k or checks whether G satisfies φ. The approach includes an O(n+m) treewidth approximation producing bounded-width decompositions, and it implies an exact treewidth computation in TLFPT time.","arXiv :2607 . 1 1230v 1 [ cs .DS] 13 Jul 2026  \nCourcelle’s Theorem in Truly Linear FPT Tuukka Korhonen∗ Daniel Lokshtanov† Saket Saurabh‡  \nAbstract  \nRecently, Bumpus, Downey, Eagling-Vose, Enright, Fellows, Kutner, Larios-Jones, Martin, Rosamond, and Yates defined Truly Linear FPT (TLFPT) to be the class of parameterized problems with algorithms running in time O (n) + f(k), where n is the input size and k the parameter [arXiv:2606.02492] . They gave several algorithmic techniques for designing TLFPT algorithms, but left parameterization by treewidth open.  \nIn this paper, we give a general method for designing TLFPT algorithms parameterized by treewidth, solving three open problems posed by Bumpus et al. In particular, we give a TLFPT algorithm for Courcelle’s theorem: We show that given an n-vertex m-edge graph G, an integer k, and a CMSO2-formula φ, we can in time O (n + m) + f(k,φ) either conclude that the treewidth of G is more than k, or check whether G satisfies φ . As a part of our algorithm, we give an approximation algorithm for treewidth that runs in time O (n + m) and returns a tree decomposition whose width is at most 2 O (k) times the optimum. Our result also implies a TLFPT algorithm for computing the value of treewidth exactly.  \n∗ University of Copenhagen, [Denmark.](Denmark. tuko@di.ku.dk. Supported)[ tuko@di.ku.dk](Denmark. tuko@di.ku.dk. Supported)[. Supported](Denmark. tuko@di.ku.dk. Supported) by the European Union under Marie Sklodowska-Curie Actions (MSCA), project no. 101206430, and by the VILLUM Foundation, Grant Number 54451, Basic Algorithms Research Copenhagen (BARC) .  \n†Department of Computer Science, University of California Santa Barbara, Santa Barbara, CA, USA. [daniello@ucsb.edu](daniello@ucsb.edu. Supported)[. Supported](daniello@ucsb.edu. Supported) by NSF Grant CCF-2505099.  \n‡The Institute of Mathematical Sciences, HBNI, Chennai, [India.](India. saket@imsc.res.in)[ saket@imsc.res.in](India. saket@imsc.res.in).  \n1 Introduction  \nIn parameterized algorithms the running time is measured in terms of the input size n and a parameter k, which captures some relevant additional information about the input instance. An algorithm is considered “good” if its running time is bounded by f (k)nc , where f is a function independent of n and c is a fixed constant independent of both n and k. Such algorithms are called fixed parameter tractable (FPT) . Not all FPT algorithms are equally good, and much research has been devoted to making f(k) grow as slowly as possible with k [CNP+22 , DFHT05 , LMS11], or c as small as possible [Bod93, IYY18 , KPS24] .  \nBeyond optimizing f(k) and nc , one could ask whether the multiplicative form of the running time is necessary, or whether one could instead obtain algorithms running in time g(k)+O(nd ) . It is well known, however, that the additive definition of FPT is equivalent to the multiplicative one: in particular g(k)+nd ≤ g(k)·nd , and similarly f(k)nc ≤ (f(k))2 + n2c. In a recent foundational paper, Bumpus et al. [BDEV+26] observed that this equivalence breaks down once one cares simultaneously about both the additive-versus-multiplicative distinction and the exponent of n. They introduced the class Truly Linear FPT (TLFPT), consisting of all problems solvable in time O (n) + f(k) for some function f, and proved, via a diagonalization argument, that there exist parameterized problems solvable in time f (k)n but not in TLFPT. This definition opens up an exciting new research direction: for problems that admit a linear FPT algorithm (one running in time f (k)n), can one obtain a TLFPT algorithm?  \nBumpus et al. [BDEV+26] gave several techniques for designing TLFPT algorithms, and posed six open problems on the existence of TLFPT algorithms for concrete problems. Three of these would be directly solved by a general method for dynamic programming parameterized by treewidth in TLFPT time. In this paper we give such a method, resolving these three questions and pa","cbCaitvOVB2SNTJE","https://ap.wps.com/l/cbCaitvOVB2SNTJE","pdf",466891,4,1,18,"English","en",105,"# Introduction\n## Truly Linear FPT and motivation\n## Main result: TLFPT for Courcelle’s theorem\n## Supporting ingredient: treewidth approximation","[{\"question\":\"What is Truly Linear FPT (TLFPT) and why does it matter in this work?\",\"answer\":\"TLFPT defines parameterized problems solvable in time O(n)+f(k). The paper focuses on whether problems with linear-in-n fixed-parameter algorithms can be transformed into TLFPT algorithms, using treewidth as the key parameter.\"},{\"question\":\"What does the paper’s main theorem guarantee for Courcelle’s theorem?\",\"answer\":\"Given an n-vertex m-edge graph G, an integer k, and a CMSO2 sentence φ, the algorithm runs in O(n+m)+f(k,φ) and either concludes tw(G)\\u003ek or determines whether G satisfies φ.\"},{\"question\":\"How is treewidth approximation used to enable the TLFPT algorithm?\",\"answer\":\"The method first computes, in O(n+m), a tree decomposition satisfying bounds on leaf and non-leaf bag sizes and the number of nodes. These properties make the decomposition suitable for dynamic programming that achieves the desired TLFPT running time.\"}]",1784212529,45,{"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},"courcelles-theorem-in-truly-linear-fpt","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/courcelles-theorem-in-truly-linear-fpt/86544/",{"url":52,"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 why does it matter in this work?","Question",{"text":75,"@type":76},"TLFPT defines parameterized problems solvable in time O(n)+f(k). The paper focuses on whether problems with linear-in-n fixed-parameter algorithms can be transformed into TLFPT algorithms, using treewidth as the key parameter.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does the paper’s main theorem guarantee for Courcelle’s theorem?",{"text":80,"@type":76},"Given an n-vertex m-edge graph G, an integer k, and a CMSO2 sentence φ, the algorithm runs in O(n+m)+f(k,φ) and either concludes tw(G)>k or determines whether G satisfies φ.",{"name":82,"@type":73,"acceptedAnswer":83},"How is treewidth approximation used to enable the TLFPT algorithm?",{"text":84,"@type":76},"The method first computes, in O(n+m), a tree decomposition satisfying bounds on leaf and non-leaf bag sizes and the number of nodes. These properties make the decomposition suitable for dynamic programming that achieves the desired TLFPT running time.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"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":20,"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"]