{"id":"https://openalex.org/W2405333517","doi":"https://doi.org/10.1017/s0963548317000049","title":"On Regularity Lemmas and their Algorithmic Applications","display_name":"On Regularity Lemmas and their Algorithmic Applications","publication_year":2017,"publication_date":"2017-03-28","ids":{"openalex":"https://openalex.org/W2405333517","doi":"https://doi.org/10.1017/s0963548317000049","mag":"2405333517"},"language":"en","primary_location":{"id":"doi:10.1017/s0963548317000049","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548317000049","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1604.00733","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5112361944","display_name":"Jacob Fox","orcid":"https://orcid.org/0000-0002-0664-497X"},"institutions":[{"id":"https://openalex.org/I97018004","display_name":"Stanford University","ror":"https://ror.org/00f54p054","country_code":"US","type":"education","lineage":["https://openalex.org/I97018004"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"JACOB FOX","raw_affiliation_strings":["Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail:"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Stanford University, Stanford, CA 94305, USA (e-mail:","institution_ids":["https://openalex.org/I97018004"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5067889485","display_name":"L\u00e1szl\u00f3 Lov\u00e1sz","orcid":"https://orcid.org/0000-0002-6242-5894"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"L\u00c1SZL\u00d3 MIKL\u00d3S LOV\u00c1SZ","raw_affiliation_strings":["Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail:"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (e-mail:","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5021133458","display_name":"Yufei Zhao","orcid":"https://orcid.org/0000-0002-1995-3755"},"institutions":[{"id":"https://openalex.org/I40120149","display_name":"University of Oxford","ror":"https://ror.org/052gg0110","country_code":"GB","type":"education","lineage":["https://openalex.org/I40120149"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"YUFEI ZHAO","raw_affiliation_strings":["Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK (e-mail:"],"affiliations":[{"raw_affiliation_string":"Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK (e-mail:","institution_ids":["https://openalex.org/I40120149"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5112361944"],"corresponding_institution_ids":["https://openalex.org/I97018004"],"apc_list":null,"apc_paid":null,"fwci":4.2641,"has_fulltext":false,"cited_by_count":22,"citation_normalized_percentile":{"value":0.94858713,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"26","issue":"4","first_page":"481","last_page":"505"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9998999834060669,"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":0.9998999834060669,"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":0.9998999834060669,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lemma","display_name":"Lemma (botany)","score":0.8464288711547852},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8168259859085083},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6710602045059204},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.6560400724411011},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.6393880248069763},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.563736617565155},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.5589922070503235},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5370259881019592},{"id":"https://openalex.org/keywords/bounding-overwatch","display_name":"Bounding overwatch","score":0.4131730794906616},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.3729197084903717}],"concepts":[{"id":"https://openalex.org/C2777759810","wikidata":"https://www.wikidata.org/wiki/Q149316","display_name":"Lemma (botany)","level":3,"score":0.8464288711547852},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8168259859085083},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6710602045059204},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.6560400724411011},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.6393880248069763},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.563736617565155},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5589922070503235},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5370259881019592},{"id":"https://openalex.org/C63584917","wikidata":"https://www.wikidata.org/wiki/Q333286","display_name":"Bounding overwatch","level":2,"score":0.4131730794906616},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.3729197084903717},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C46757340","wikidata":"https://www.wikidata.org/wiki/Q43238","display_name":"Poaceae","level":2,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1017/s0963548317000049","is_oa":false,"landing_page_url":"https://doi.org/10.1017/s0963548317000049","pdf_url":null,"source":{"id":"https://openalex.org/S7291079","display_name":"Combinatorics Probability Computing","issn_l":"0963-5483","issn":["0963-5483","1469-2163"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Combinatorics, Probability and Computing","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1604.00733","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.00733","pdf_url":"https://arxiv.org/pdf/1604.00733","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1604.00733","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1604.00733","pdf_url":"https://arxiv.org/pdf/1604.00733","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W273178449","https://openalex.org/W648165475","https://openalex.org/W648608585","https://openalex.org/W1691611448","https://openalex.org/W1780823682","https://openalex.org/W1960897633","https://openalex.org/W1979740015","https://openalex.org/W1997230618","https://openalex.org/W2004694806","https://openalex.org/W2019367452","https://openalex.org/W2052318581","https://openalex.org/W2068571612","https://openalex.org/W2068871408","https://openalex.org/W2072858942","https://openalex.org/W2081254453","https://openalex.org/W2083580384","https://openalex.org/W2092086436","https://openalex.org/W2111754130","https://openalex.org/W2120248756","https://openalex.org/W2124366921","https://openalex.org/W2126186592","https://openalex.org/W2128139967","https://openalex.org/W2141649469","https://openalex.org/W2141765721","https://openalex.org/W2160594140","https://openalex.org/W2293834400","https://openalex.org/W2401939847","https://openalex.org/W2405333517","https://openalex.org/W2610183793","https://openalex.org/W2977322974","https://openalex.org/W3023108254","https://openalex.org/W3104055895"],"related_works":["https://openalex.org/W3006682748","https://openalex.org/W4254119641","https://openalex.org/W3196207352","https://openalex.org/W2951724202","https://openalex.org/W3084261076","https://openalex.org/W2576399385","https://openalex.org/W2963408011","https://openalex.org/W1998171255","https://openalex.org/W4221036368","https://openalex.org/W2044959892"],"abstract_inverted_index":{"Szemer\u00e9di's":[0],"regularity":[1,24,44,70,118,135,140,181],"lemma":[2,45,71,136,182],"and":[3,101],"its":[4],"variants":[5],"are":[6],"some":[7],"of":[8,55,57,66,114,141,144,178,191],"the":[9,23,35,38,43,53,58,67,92,96,112,132,139,179,185,189,194],"most":[10],"powerful":[11],"tools":[12],"in":[13,42,83,193],"combinatorics.":[14],"In":[15],"this":[16],"paper,":[17],"we":[18,27,33,61,102,110,173],"establish":[19],"several":[20,124],"results":[21],"around":[22],"lemma.":[25],"First,":[26],"prove":[28],"that":[29,37],"whether":[30],"or":[31],"not":[32],"include":[34],"condition":[36],"desired":[39],"vertex":[40,145],"partition":[41,159,168],"is":[46],"equitable":[47],"has":[48],"a":[49,74,84,106,142,175,198],"minimal":[50],"effect":[51],"on":[52,95,188],"number":[54,190],"parts":[56,162,192],"partition.":[59],"Second,":[60],"use":[62,131],"an":[63,88,116,157,166],"algorithmic":[64,117],"version":[65],"(weak)":[68],"Frieze\u2013Kannan":[69,134],"to":[72,105,130,137,151,197],"give":[73,174],"substantially":[75],"faster":[76],"deterministic":[77],"approximation":[78,121],"algorithm":[79],"for":[80,91,123,154],"counting":[81],"subgraphs":[82],"graph.":[85],"Previously,":[86],"only":[87],"exponential":[89,200],"dependence":[90],"running":[93],"time":[94],"error":[97],"parameter":[98],"was":[99],"known,":[100],"improve":[103],"it":[104],"polynomial":[107],"dependence.":[108],"Third,":[109],"revisit":[111],"problem":[113],"finding":[115],"lemma,":[119],"giving":[120],"algorithms":[122],"co-NP-complete":[125],"problems.":[126],"We":[127,147],"show":[128,149],"how":[129,150],"weak":[133],"approximate":[138],"pair":[143],"subsets.":[146],"also":[148],"quickly":[152],"find,":[153],"each":[155],"\u03b5\u2032>\u03b5,":[156],"\u03b5\u2032-regular":[158],"with":[160,169],"k":[161,170],"if":[163],"there":[164],"exists":[165],"\u03b5-regular":[167],"parts.":[171],"Finally,":[172],"simple":[176],"proof":[177],"permutation":[180],"which":[183],"improves":[184],"tower-type":[186],"bound":[187],"previous":[195],"proofs":[196],"single":[199],"bound.":[201]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":2}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}