{"id":"https://openalex.org/W3043603630","doi":"https://doi.org/10.1137/18m1176701","title":"On Approximating the Number of $k$-Cliques in Sublinear Time","display_name":"On Approximating the Number of $k$-Cliques in Sublinear Time","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3043603630","doi":"https://doi.org/10.1137/18m1176701","mag":"3043603630"},"language":"en","primary_location":{"id":"doi:10.1137/18m1176701","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1176701","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5086741302","display_name":"Talya Eden","orcid":"https://orcid.org/0000-0001-8470-9508"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Talya Eden","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5063746987","display_name":"Dana Ron","orcid":"https://orcid.org/0000-0001-6576-7200"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Dana Ron","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5038687804","display_name":"C. Seshadhri","orcid":"https://orcid.org/0000-0003-2163-3555"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"C. Seshadhri","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":1.3813,"has_fulltext":false,"cited_by_count":13,"citation_normalized_percentile":{"value":0.85608696,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"49","issue":"4","first_page":"747","last_page":"771"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.9998000264167786,"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/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.9966999888420105,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8003066778182983},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7004817724227905},{"id":"https://openalex.org/keywords/sublinear-function","display_name":"Sublinear function","score":0.6021050214767456},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5256242156028748},{"id":"https://openalex.org/keywords/binary-logarithm","display_name":"Binary logarithm","score":0.5030466914176941},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.46517372131347656},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.4486866891384125},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.447734534740448},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.43189537525177},{"id":"https://openalex.org/keywords/running-time","display_name":"Running time","score":0.41011521220207214},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.2084236741065979}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8003066778182983},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7004817724227905},{"id":"https://openalex.org/C117160843","wikidata":"https://www.wikidata.org/wiki/Q338652","display_name":"Sublinear function","level":2,"score":0.6021050214767456},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5256242156028748},{"id":"https://openalex.org/C63553672","wikidata":"https://www.wikidata.org/wiki/Q581168","display_name":"Binary logarithm","level":2,"score":0.5030466914176941},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.46517372131347656},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.4486866891384125},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.447734534740448},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.43189537525177},{"id":"https://openalex.org/C3017489831","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Running time","level":2,"score":0.41011521220207214},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2084236741065979},{"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/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/18m1176701","is_oa":false,"landing_page_url":"https://doi.org/10.1137/18m1176701","pdf_url":null,"source":{"id":"https://openalex.org/S153560523","display_name":"SIAM Journal on Computing","issn_l":"0097-5397","issn":["0097-5397","1095-7111"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G440923042","display_name":null,"funder_award_id":"671/13","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"},{"id":"https://openalex.org/G6330059706","display_name":null,"funder_award_id":"1146/18","funder_id":"https://openalex.org/F4320322252","funder_display_name":"Israel Science Foundation"},{"id":"https://openalex.org/G7685655407","display_name":"TRIPODS: Institute for Foundations of Data Science (IFDS)","funder_award_id":"1740751","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320312914","display_name":"Blavatnik Family Foundation","ror":null},{"id":"https://openalex.org/F4320322252","display_name":"Israel Science Foundation","ror":"https://ror.org/04sazxf24"},{"id":"https://openalex.org/F4320323139","display_name":"Azrieli Foundation","ror":"https://ror.org/009svts64"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W1485742133","https://openalex.org/W1489509891","https://openalex.org/W1542957328","https://openalex.org/W1866637071","https://openalex.org/W1963750764","https://openalex.org/W1967022297","https://openalex.org/W1973113631","https://openalex.org/W1980155175","https://openalex.org/W1985339741","https://openalex.org/W1994987445","https://openalex.org/W2001342485","https://openalex.org/W2002779084","https://openalex.org/W2024859348","https://openalex.org/W2042587503","https://openalex.org/W2055245094","https://openalex.org/W2065221212","https://openalex.org/W2074724691","https://openalex.org/W2074757518","https://openalex.org/W2081449822","https://openalex.org/W2082227928","https://openalex.org/W2116759966","https://openalex.org/W2117071406","https://openalex.org/W2153624566","https://openalex.org/W2155978818","https://openalex.org/W2761580201","https://openalex.org/W3021019285","https://openalex.org/W3090803701","https://openalex.org/W3123534837","https://openalex.org/W3124674038","https://openalex.org/W3126028654","https://openalex.org/W3160042174","https://openalex.org/W4230053343","https://openalex.org/W4292080745"],"related_works":["https://openalex.org/W3127082728","https://openalex.org/W1983915440","https://openalex.org/W2067936666","https://openalex.org/W3095925971","https://openalex.org/W2113270940","https://openalex.org/W2004283824","https://openalex.org/W1581210843","https://openalex.org/W1992104669","https://openalex.org/W1982530285","https://openalex.org/W2521054645"],"abstract_inverted_index":{"We":[0,20,61,215],"study":[1],"the":[2,6,18,22,43,48,51,57,87,90,95,98,112,123,184,238],"problem":[3],"of":[4,8,45,53,59,89,97,115,186,237,240,245],"approximating":[5],"number":[7,44,52,58],"$k$-cliques":[9],"in":[10,47,94,134],"a":[11,67,107,198,217,228,241],"graph":[12,99],"when":[13],"given":[14,242],"query":[15,24,76,113],"access":[16],"to":[17,122,235],"graph.":[19],"consider":[21],"standard":[23],"model":[25],"for":[26,72,100,159,177,205,212,222],"general":[27,218],"graphs":[28],"via":[29],"(1)":[30],"degree":[31],"queries,":[32,35],"(2)":[33],"neighbor":[34],"and":[36,55,78,129,147,150,163,168,208],"(3)":[37],"pair":[38],"queries.":[39],"Let":[40],"$n$":[41],"denote":[42],"vertices":[46,239,246],"graph,":[49],"$m$":[50],"edges,":[54],"$C_k$":[56],"$k$-cliques.":[60],"design":[62],"an":[63],"algorithm":[64,91,117,219],"that":[65,111,202,220,230],"outputs":[66],"$(1+\\varepsilon)$-approximation":[68],"(with":[69],"high":[70],"probability)":[71],"$C_k$,":[73],"whose":[74],"expected":[75],"complexity":[77,88,114],"running":[79],"time":[80],"are":[81,137],"$O(\\frac{n}{C_k^{1/k}}+\\frac{m^{k/2}}{C_k}":[82],"){poly}(\\log":[83],"n,":[84],"1/\\varepsilon,k)$.":[85],"Hence,":[86],"is":[92,118],"sublinear":[93],"size":[96],"$C_k":[101],"=":[102],"\\omega(m^{k/2-1})$.":[103],"Furthermore,":[104],"we":[105],"prove":[106],"lower":[108],"bound":[109],"showing":[110],"our":[116],"essentially":[119],"optimal":[120],"(up":[121],"dependence":[124],"on":[125,197],"$\\log":[126],"n$,":[127],"$1/\\varepsilon$,":[128],"$k$).":[130],"The":[131,189],"previous":[132,190],"results":[133],"this":[135],"vein":[136],"by":[138,148,164,192,226],"Feige":[139],"[SIAM":[140,170],"J.":[141,171],"Comput.,":[142,172],"35":[143],"(2006),":[144],"pp.":[145,157,175],"964--984]":[146],"Goldreich":[149],"Ron":[151],"[Random":[152],"Structures":[153],"Algorithms,":[154],"32":[155],"(2008),":[156],"473--493]":[158],"edge":[160],"counting":[161,179,207],"($k=2$)":[162],"Eden,":[165],"Levi,":[166],"Ron,":[167],"Seshadhri":[169],"46":[173],"(2017),":[174],"1603--1646]":[176],"triangle":[178,206],"($k=3$).":[180],"Our":[181],"result":[182,191],"matches":[183],"complexities":[185],"these":[187],"results.":[188],"Eden":[193],"et":[194],"al.":[195],"hinges":[196],"certain":[199],"amortization":[200],"technique":[201],"works":[203,221],"only":[204],"does":[209],"not":[210],"generalize":[211],"larger":[213],"cliques.":[214],"obtain":[216],"any":[223],"$k\\geq":[224],"3$":[225],"designing":[227],"procedure":[229],"samples":[231],"each":[232],"$k$-clique":[233],"incident":[234],"one":[236],"set":[243],"$S$":[244],"with":[247],"approximately":[248],"equal":[249],"probability.":[250]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
