{"id":"https://openalex.org/W7163196246","doi":"https://doi.org/10.48550/arxiv.2606.02492","title":"$O(n +f(k))$: Truly Linear FPT","display_name":"$O(n +f(k))$: Truly Linear FPT","publication_year":2026,"publication_date":"2026-06-01","ids":{"openalex":"https://openalex.org/W7163196246","doi":"https://doi.org/10.48550/arxiv.2606.02492"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2606.02492","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.02492","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2606.02492","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020233690","display_name":"Benjamin Merlin Bumpus","orcid":"https://orcid.org/0000-0002-8686-2319"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bumpus, Benjamin Merlin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5110868906","display_name":"Rod Downey","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Downey, Rod","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5094063465","display_name":"Tala Eagling-Vose","orcid":"https://orcid.org/0009-0008-0346-7032"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Eagling-Vose, Tala","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5136502737","display_name":"Jessica Enright","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Enright, Jessica","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137707743","display_name":"Michael R. Fellows","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Fellows, Michael R.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086433219","display_name":"David C. Kutner","orcid":"https://orcid.org/0000-0003-2979-4513"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kutner, David C.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5092160873","display_name":"Laura Larios-Jones","orcid":"https://orcid.org/0000-0003-3322-0176"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Larios-Jones, Laura","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137707035","display_name":"Barnaby Martin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Martin, Barnaby","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5137671774","display_name":"Frances Rosamond","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Rosamond, Frances","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5119872371","display_name":"Ella Yates","orcid":"https://orcid.org/0009-0000-7979-0401"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yates, Ella","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.6758999824523926,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.6758999824523926,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.09369999915361404,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12292","display_name":"Graph Theory and Algorithms","score":0.02710000053048134,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.9031000137329102},{"id":"https://openalex.org/keywords/leverage","display_name":"Leverage (statistics)","score":0.6309000253677368},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.4796999990940094},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.4636000096797943},{"id":"https://openalex.org/keywords/heuristics","display_name":"Heuristics","score":0.4629000127315521},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.45190000534057617},{"id":"https://openalex.org/keywords/vertex-cover","display_name":"Vertex cover","score":0.4325000047683716},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.3928999900817871}],"concepts":[{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.9031000137329102},{"id":"https://openalex.org/C153083717","wikidata":"https://www.wikidata.org/wiki/Q6535263","display_name":"Leverage (statistics)","level":2,"score":0.6309000253677368},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5015000104904175},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.4796999990940094},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.4636000096797943},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.4629000127315521},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.45190000534057617},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43689998984336853},{"id":"https://openalex.org/C40687702","wikidata":"https://www.wikidata.org/wiki/Q11515519","display_name":"Vertex cover","level":3,"score":0.4325000047683716},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.42719998955726624},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.3928999900817871},{"id":"https://openalex.org/C34628019","wikidata":"https://www.wikidata.org/wiki/Q13636890","display_name":"Algorithmics","level":2,"score":0.3716000020503998},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3562000095844269},{"id":"https://openalex.org/C2778755073","wikidata":"https://www.wikidata.org/wiki/Q10858537","display_name":"Scale (ratio)","level":2,"score":0.32839998602867126},{"id":"https://openalex.org/C179799912","wikidata":"https://www.wikidata.org/wiki/Q205084","display_name":"Computational complexity theory","level":2,"score":0.3228999972343445},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3107999861240387},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.30090001225471497},{"id":"https://openalex.org/C75684735","wikidata":"https://www.wikidata.org/wiki/Q858810","display_name":"Big data","level":2,"score":0.2971000075340271},{"id":"https://openalex.org/C115908005","wikidata":"https://www.wikidata.org/wiki/Q2668364","display_name":"Combinatorial explosion","level":2,"score":0.2935999929904938},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.289900004863739},{"id":"https://openalex.org/C148764684","wikidata":"https://www.wikidata.org/wiki/Q621751","display_name":"Approximation algorithm","level":2,"score":0.2874999940395355},{"id":"https://openalex.org/C41045048","wikidata":"https://www.wikidata.org/wiki/Q202843","display_name":"Linear programming","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C207225210","wikidata":"https://www.wikidata.org/wiki/Q1759539","display_name":"Kernelization","level":3,"score":0.2540999948978424}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2606.02492","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.02492","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2606.02492","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.02492","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Parameterized":[0],"complexity":[1,71,216],"has":[2,38,66],"always":[3],"been":[4],"concerned":[5],"with":[6,229],"practical":[7,64,246],"computing:":[8],"by":[9,205],"confining":[10],"combinatorial":[11],"explosion":[12],"to":[13,74,198,217,221,227],"a":[14,63,130,245,256],"secondary":[15],"parameter":[16,80],"$k$,":[17],"one":[18,100],"can":[19],"uncover":[20],"why":[21],"and":[22,101,114,125,154,174,192,211,247],"how":[23],"many":[24,159],"NP-hard":[25],"problems":[26],"are":[27,178,195],"effectively":[28],"tackled":[29],"in":[30,52],"practice.":[31],"Today,":[32],"however,":[33],"the":[34,53,75,96,199,222],"scale":[35],"of":[36,133,151,161,171,219],"data":[37,155],"changed:":[39],"scientists":[40],"study":[41],"Big":[42,230],"Data,":[43],"which":[44,194],"is":[45,58,72,99,106,122,129,139,244,255],"so":[46,204],"large":[47,234],"that":[48,121,127,138],"even":[49],"quadratic":[50],"dependence":[51],"total":[54],"input":[55],"size":[56],"$n$":[57],"unaffordable.":[59],"Therefore,":[60],"what":[61,94],"constitutes":[62],"algorithm":[65],"also":[67],"changed.":[68],"Classically,":[69],"parameterized":[70,215],"blind":[73],"difference":[76],"between":[77],"defining":[78],"fixed":[79],"tractability":[81],"multiplicatively":[82],"(i.e.":[83,89],"$f(k)":[84,90],"\\cdot":[85,141],"n^c$)":[86],"or":[87],"additively":[88],"+":[91],"n^c$).":[92],"But":[93],"if":[95],"constant":[97],"$c$":[98],"we":[102,112,187,225],"require":[103],"true":[104],"linearity,":[105],"this":[107],"distinction":[108],"still":[109],"inconsequential?":[110],"Here,":[111],"define":[113],"explore":[115],"Truly":[116],"Linear":[117,134],"FPT":[118,135,236],"(TLFPT)":[119],"--":[120,124,137,143],"$O(n)+f(k)$":[123],"show":[126],"it":[128],"strict":[131],"subset":[132],"(LFPT)":[136],"$O(n)":[140],"f(k)$":[142],"via":[144],"diagonalization.":[145],"Populating":[146],"TLFPT":[147,200,254],"requires":[148],"careful":[149],"consideration":[150],"linear-time":[152],"algorithmics":[153],"structures.":[156],"We":[157,202],"meet":[158],"inhabitants":[160],"TLFPT:":[162],"SAT,":[163],"Vertex":[164],"Cover,":[165],"Min-Max":[166],"Matching,":[167],"$(n-k)$-Coloring,":[168],"Diverse":[169],"Pair":[170],"Matchings,":[172],"$k$-Path,":[173],"$H$-Coloring.":[175],"Our":[176],"parameterizations":[177],"equally":[179],"varied.":[180],"Beyond":[181],"classical":[182],"parameters":[183],"like":[184],"solution":[185],"size,":[186],"leverage":[188],"two":[189],"parameters,":[190],"treedepth":[191],"BFS-width,":[193],"particularly":[196],"well-suited":[197],"regime.":[201],"do":[203],"developing":[206],"techniques":[207],"based":[208],"on":[209],"depth-":[210],"breadth-first":[212],"search.":[213],"For":[214,232],"be":[218],"service":[220],"scientific":[223],"community,":[224],"need":[226,249],"contend":[228],"Data.":[231],"sufficiently":[233],"inputs,":[235],"beyond":[237],"linear":[238],"may":[239],"not":[240],"suffice.":[241],"Thus,":[242],"there":[243],"theoretical":[248],"for":[250],"more":[251],"ambitious":[252],"goals.":[253],"first":[257],"step":[258],"forward.":[259]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-06-03T00:00:00"}
