{"id":"https://openalex.org/W646298202","doi":null,"title":"Convex relaxation for the planted clique, biclique, and clustering problems","display_name":"Convex relaxation for the planted clique, biclique, and clustering problems","publication_year":2011,"publication_date":"2011-08-24","ids":{"openalex":"https://openalex.org/W646298202","mag":"646298202"},"language":"en","primary_location":{"id":"pmh:oai:uwspace.uwaterloo.ca:10012/6113","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10012/6113","pdf_url":"http://hdl.handle.net/10012/6113","source":{"id":"https://openalex.org/S4306401661","display_name":"UWSpace (University of Waterloo)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I151746483","host_organization_name":"University of Waterloo","host_organization_lineage":["https://openalex.org/I151746483"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Doctoral Thesis"},"volume":null,"type":"dissertation","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/10012/6113","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5000908908","display_name":"Brendan Ames","orcid":"https://orcid.org/0000-0003-0810-7956","observed_orcids":["https://orcid.org/0000-0003-0810-7956"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Brendan Ames","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0003-0810-7956","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5000908908"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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/T10720","display_name":"Complexity and Algorithms in Graphs","score":0.8105000257492065,"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":0.8105000257492065,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.0471000000834465,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.04500000178813934,"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/planted-clique","display_name":"planted clique","score":0.6729999780654907},{"id":"https://openalex.org/keywords/maximum-clique-problem","display_name":"maximum clique problem","score":0.8130000233650208},{"id":"https://openalex.org/keywords/complete-bipartite-graphs","display_name":"complete bipartite graphs","score":0.7049999833106995},{"id":"https://openalex.org/keywords/clustering","display_name":"clustering","score":0.460999995470047},{"id":"https://openalex.org/keywords/nuclear-norm-relaxation","display_name":"nuclear norm relaxation","score":0.7519999742507935},{"id":"https://openalex.org/keywords/rank-minimization","display_name":"rank minimization","score":0.9049999713897705},{"id":"https://openalex.org/keywords/convex-relaxation","display_name":"convex relaxation","score":0.7300000190734863},{"id":"https://openalex.org/keywords/random-graphs","display_name":"random graphs","score":0.7099999785423279},{"id":"https://openalex.org/keywords/graph-partitioning","display_name":"graph partitioning","score":0.48399999737739563},{"id":"https://openalex.org/keywords/np-hardness","display_name":"NP-hardness","score":0.5270000100135803},{"id":"https://openalex.org/keywords/combinatorial-optimization","display_name":"combinatorial optimization","score":0.5},{"id":"https://openalex.org/keywords/semidefinite-programming","display_name":"semidefinite programming","score":0.4869999885559082},{"id":"https://openalex.org/keywords/spectral-partitioning","display_name":"spectral partitioning","score":0.34599998593330383},{"id":"https://openalex.org/keywords/graph-sparsification","display_name":"graph sparsification","score":0.3089999854564667}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8075032234191895},{"id":"https://openalex.org/C134119311","wikidata":"https://www.wikidata.org/wiki/Q913598","display_name":"Complete bipartite graph","level":4,"score":0.7696470022201538},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.691635251045227},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.6180947422981262},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.5117645263671875},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.4552915692329407},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.44734781980514526},{"id":"https://openalex.org/C150997102","wikidata":"https://www.wikidata.org/wiki/Q1196873","display_name":"Clique problem","level":5,"score":0.42939549684524536},{"id":"https://openalex.org/C189239283","wikidata":"https://www.wikidata.org/wiki/Q17101022","display_name":"Clique graph","level":5,"score":0.4108904004096985},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2742200493812561},{"id":"https://openalex.org/C149530733","wikidata":"https://www.wikidata.org/wiki/Q5597091","display_name":"Graph power","level":4,"score":0.20700109004974365},{"id":"https://openalex.org/C203776342","wikidata":"https://www.wikidata.org/wiki/Q1378376","display_name":"Line graph","level":3,"score":0.18967187404632568},{"id":"https://openalex.org/C43517604","wikidata":"https://www.wikidata.org/wiki/Q7144893","display_name":"Pathwidth","level":4,"score":0.09833446145057678}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:uwspace.uwaterloo.ca:10012/6113","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10012/6113","pdf_url":"http://hdl.handle.net/10012/6113","source":{"id":"https://openalex.org/S4306401661","display_name":"UWSpace (University of Waterloo)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I151746483","host_organization_name":"University of Waterloo","host_organization_lineage":["https://openalex.org/I151746483"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Doctoral Thesis"},{"id":"mag:646298202","doi":null,"is_oa":false,"landing_page_url":"https://uwspace.uwaterloo.ca/bitstream/10012/6113/1/Ames_Brendan.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:uwspace.uwaterloo.ca:10012/6113","doi":null,"is_oa":true,"landing_page_url":"http://hdl.handle.net/10012/6113","pdf_url":"http://hdl.handle.net/10012/6113","source":{"id":"https://openalex.org/S4306401661","display_name":"UWSpace (University of Waterloo)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I151746483","host_organization_name":"University of Waterloo","host_organization_lineage":["https://openalex.org/I151746483"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Doctoral Thesis"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320322676","display_name":"University of Waterloo","ror":"https://ror.org/01aff2v68"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W646298202.pdf","grobid_xml":"https://content.openalex.org/works/W646298202.grobid-xml"},"referenced_works_count":180,"referenced_works":["https://openalex.org/W51435562","https://openalex.org/W106934005","https://openalex.org/W168026768","https://openalex.org/W177258377","https://openalex.org/W391578156","https://openalex.org/W604404646","https://openalex.org/W641613623","https://openalex.org/W1481481825","https://openalex.org/W1487444668","https://openalex.org/W1488435683","https://openalex.org/W1501500081","https://openalex.org/W1511814458","https://openalex.org/W1529624360","https://openalex.org/W1539402525","https://openalex.org/W1550538040","https://openalex.org/W1550611075","https://openalex.org/W1555834419","https://openalex.org/W1573301526","https://openalex.org/W1575824241","https://openalex.org/W1587744656","https://openalex.org/W1603194053","https://openalex.org/W1605711022","https://openalex.org/W1632601927","https://openalex.org/W1771423625","https://openalex.org/W1936234780","https://openalex.org/W1964089073","https://openalex.org/W1964119857","https://openalex.org/W1964730816","https://openalex.org/W1966096622","https://openalex.org/W1974213704","https://openalex.org/W1976618413","https://openalex.org/W1976713424","https://openalex.org/W1977556410","https://openalex.org/W1978578701","https://openalex.org/W1984217603","https://openalex.org/W1985123706","https://openalex.org/W1986007546","https://openalex.org/W1986219860","https://openalex.org/W1986931325","https://openalex.org/W1989720410","https://openalex.org/W1996287810","https://openalex.org/W1999913571","https://openalex.org/W2003753589","https://openalex.org/W2004544971","https://openalex.org/W2005412366","https://openalex.org/W2007437396","https://openalex.org/W2007940874","https://openalex.org/W2009006054","https://openalex.org/W2009394266","https://openalex.org/W2010601719","https://openalex.org/W2011039300","https://openalex.org/W2011359124","https://openalex.org/W2012961725","https://openalex.org/W2013568176","https://openalex.org/W2015932681","https://openalex.org/W2019578639","https://openalex.org/W2020567423","https://openalex.org/W2025666718","https://openalex.org/W2026320281","https://openalex.org/W2028126706","https://openalex.org/W2028781966","https://openalex.org/W2029213856","https://openalex.org/W2033639703","https://openalex.org/W2033852356","https://openalex.org/W2034331023","https://openalex.org/W2039778616","https://openalex.org/W2040924621","https://openalex.org/W2041588219","https://openalex.org/W2042463931","https://openalex.org/W2043005464","https://openalex.org/W2045027193","https://openalex.org/W2046701389","https://openalex.org/W2048440507","https://openalex.org/W2050834445","https://openalex.org/W2051006653","https://openalex.org/W2055107590","https://openalex.org/W2056042427","https://openalex.org/W2056305270","https://openalex.org/W2062307640","https://openalex.org/W2066459185","https://openalex.org/W2068871408","https://openalex.org/W2072439030","https://openalex.org/W2073025989","https://openalex.org/W2074471762","https://openalex.org/W2075894130","https://openalex.org/W2078289947","https://openalex.org/W2078758709","https://openalex.org/W2081254453","https://openalex.org/W2081323189","https://openalex.org/W2082603400","https://openalex.org/W2083694956","https://openalex.org/W2086653003","https://openalex.org/W2087962968","https://openalex.org/W2088164510","https://openalex.org/W2089247135","https://openalex.org/W2097844290","https://openalex.org/W2099168123","https://openalex.org/W2100440346","https://openalex.org/W2100505346","https://openalex.org/W2100556411","https://openalex.org/W2102823806","https://openalex.org/W2103012681","https://openalex.org/W2103972604","https://openalex.org/W2107940287","https://openalex.org/W2109407305","https://openalex.org/W2109449402","https://openalex.org/W2111826413","https://openalex.org/W2113740359","https://openalex.org/W2114030927","https://openalex.org/W2116007667","https://openalex.org/W2116616571","https://openalex.org/W2117065890","https://openalex.org/W2117513548","https://openalex.org/W2117942072","https://openalex.org/W2118550318","https://openalex.org/W2120872934","https://openalex.org/W2121947440","https://openalex.org/W2125278290","https://openalex.org/W2125531986","https://openalex.org/W2127450210","https://openalex.org/W2129131372","https://openalex.org/W2129133480","https://openalex.org/W2129638195","https://openalex.org/W2132603077","https://openalex.org/W2134332047","https://openalex.org/W2134737843","https://openalex.org/W2135127969","https://openalex.org/W2135957668","https://openalex.org/W2136120429","https://openalex.org/W2141815340","https://openalex.org/W2143391442","https://openalex.org/W2144123706","https://openalex.org/W2144730813","https://openalex.org/W2145962650","https://openalex.org/W2147737043","https://openalex.org/W2150698455","https://openalex.org/W2150718242","https://openalex.org/W2158923808","https://openalex.org/W2160090507","https://openalex.org/W2160429376","https://openalex.org/W2161675672","https://openalex.org/W2162148987","https://openalex.org/W2162451874","https://openalex.org/W2162581541","https://openalex.org/W2164452299","https://openalex.org/W2165107146","https://openalex.org/W2165916500","https://openalex.org/W2166163936","https://openalex.org/W2166602562","https://openalex.org/W2167077875","https://openalex.org/W2167649597","https://openalex.org/W2167732364","https://openalex.org/W2168745297","https://openalex.org/W2176446742","https://openalex.org/W2183845893","https://openalex.org/W2296319761","https://openalex.org/W2296616510","https://openalex.org/W2339666411","https://openalex.org/W2341535507","https://openalex.org/W2398665108","https://openalex.org/W2401610261","https://openalex.org/W2550766755","https://openalex.org/W2611328865","https://openalex.org/W2614891641","https://openalex.org/W2616032753","https://openalex.org/W2797618585","https://openalex.org/W2798707604","https://openalex.org/W2798909945","https://openalex.org/W2899702797","https://openalex.org/W2912522929","https://openalex.org/W2913066018","https://openalex.org/W2951927428","https://openalex.org/W3022380717","https://openalex.org/W3029645440","https://openalex.org/W3102105495","https://openalex.org/W3104501706","https://openalex.org/W3138835833","https://openalex.org/W3139328003","https://openalex.org/W4302155765","https://openalex.org/W7030295306"],"related_works":["https://openalex.org/W7098144578","https://openalex.org/W2116616571","https://openalex.org/W7099918456","https://openalex.org/W1936234780","https://openalex.org/W4234658180","https://openalex.org/W944123","https://openalex.org/W1572312821","https://openalex.org/W4247854967","https://openalex.org/W7098159605","https://openalex.org/W4384807922"],"abstract_inverted_index":{"A":[0],"clique":[1,57,65,99,170,245],"of":[2,9,13,21,29,53,58,74,102,121,185,231,268,272,281,303,307,313,317,324,328,385,443,464,486,500,525,532,535],"a":[3,7,16,22,27,59,75,98,103,107,113,118,168,266,273,340,353,388,424,431,498],"graph":[4,24,77,108,166,213,276,285,379,426],"G":[5,25,286,308,329,380,418],"is":[6,26,40,109,158,189,209,263,414,513],"set":[8,120,267,302,356,384,391,434],"pairwise":[10],"adjacent":[11,41,397],"nodes":[12,175,386,395,428],"G.":[14,48],"Similarly,":[15,422],"biclique":[17,70,92,101,172],"(U,":[18,71],"V":[19,46,72],")":[20,73],"bipartite":[23,76],"pair":[28,442],"disjoint,":[30],"independent":[31],"vertex":[32],"sets":[33,546],"such":[34,381],"that":[35,78,96,155,163,309,358,382,461,491,522],"each":[36,179,206,363,441],"node":[37,44,260],"in":[38,45,47,106,160,362,372],"U":[39],"to":[42,111,135,211,254,264,299,322,342,351,416,430,478,497,538],"every":[43],"We":[49,94,488,518],"consider":[50],"the":[51,55,63,69,80,89,145,161,164,192,204,212,222,228,243,255,278,292,301,311,314,318,325,332,344,359,370,378,383,402,408,445,452,465,482,493],"problems":[52,125,132,149,335,475],"identifying":[54,68],"maximum":[56,64,90,244,259,279,293,453],"graph,":[60],"known":[61,87],"as":[62,88,129,236,238,339,471,542,544],"problem,":[66,180],"and":[67,133,176,234,240,252,287,369,392,399,435,476,508,528],"maximizes":[79,310],"product":[81],"|U":[82],"|":[83],"\u00b7":[84],"|V":[85],"|,":[86],"edge":[91,208,289,436],"problem.":[93,246,257,346,457],"show":[95,154,460,490],"\ufb01nding":[97,112],"or":[100,171,365],"given":[104,274,354,389,423,432],"size":[105],"equivalent":[110,415],"rank":[114,130,140,472],"one":[115,349],"matrix":[116],"satisfying":[117],"particular":[119],"linear":[122],"constraints.":[123],"These":[124,334],"can":[126,468],"be":[127,337,448,469],"formulated":[128,470],"minimization":[131],"relaxed":[134,477],"convex":[136,143],"programming":[137],"by":[138,200,331,450],"replacing":[139],"with":[141,217,320,427],"its":[142],"envelope,":[144],"nuclear":[146,483],"norm.":[147],"Both":[148],"are":[150,197,367,375,396,405],"NP-hard":[151],"yet":[152],"we":[153,181],"our":[156,187,225,526,536],"relaxation":[157,188,485,512,527],"exact":[159,514],"case":[162],"input":[165,275,284,494],"contains":[167],"large":[169],"plus":[173,505],"additional":[174,506],"edges.":[177],"For":[178,377],"provide":[182,519,529],"two":[183,394],"analyses":[184],"when":[186,492],"exact.":[190],"In":[191,203,221,347],"\ufb01rst,":[193],"\\nthe":[194],"diversionary":[195],"edges":[196,507],"added":[198,210],"deterministically":[199],"an":[201],"adversary.":[202],"second,":[205],"potential":[207],"independently":[214],"at":[215],"random":[216,223],"\ufb01xed":[218],"probability":[219],"p.":[220],"case,":[224],"bounds":[226,230,521],"match":[227],"earlier":[229],"Alon,":[232],"Krivelevich,":[233],"Sudakov,":[235],"well":[237,543],"Feige":[239],"Krauthgamer":[241],"for":[242,515],"\\n":[247,458],"\\nWe":[248,459],"extend":[249],"these":[250],"results":[251],"techniques":[253],"k-disjoint-clique":[256,261,296,456,466],"The":[258],"problem":[262,297,467],"\ufb01nd":[265],"k":[269,304,411,501],"disjoint":[270,305,412,502],"cliques":[271,306,504],"containing":[277],"number":[280],"nodes.":[282],"Given":[283],"nonnegative":[288],"\\nweights":[290],"w,":[291,323],"mean":[294,454],"weight":[295,455],"seeks":[298],"identify":[300],"sum":[312],"average":[315],"weights":[316,437],"edges,":[319],"respect":[321],"complete":[326,425],"subgraphs":[327],"induced":[330],"cliques.":[333,421],"may":[336,447],"considered":[338],"way":[341],"pose":[343],"clustering":[345,407,548],"clustering,":[348],"wants":[350],"partition":[352,364],"data":[355,360,390,409,433,446,540,545],"so":[357],"items":[361,371,404],"cluster":[366],"similar":[368],"di\ufb00erent":[373],"clusters":[374,413],"dissimilar.":[376],"represents":[387],"any":[393],"if":[398,401],"only":[400],"corresponding":[403,429],"similar,":[406],"into":[410,419],"partitioning":[417],"k-disjoint":[420],"indicating":[438],"similarity":[439],"between":[440],"items,":[444],"clustered":[449],"solving":[451],"both":[462,516],"instances":[463],"constrained":[473],"optimization":[474],"semide\ufb01nite":[479,511],"programs":[480],"using":[481],"norm":[484],"rank.":[487],"also":[489],"instance":[495],"corresponds":[496],"collection":[499],"planted":[503],"nodes,":[509],"this":[510],"problems.":[517],"theoretical":[520],"guarantee":[523],"exactness":[524],"empirical":[530],"examples":[531],"successful":[533],"applications":[534],"algorithm":[537],"synthetic":[539],"sets,":[541],"from":[547],"applications.":[549]},"counts_by_year":[{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2016-06-24T00:00:00"}