{"id":"https://openalex.org/W2103718624","doi":"https://doi.org/10.1007/978-3-642-00202-1_24","title":"The Planar k-Means Problem is NP-Hard","display_name":"The Planar k-Means Problem is NP-Hard","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2103718624","doi":"https://doi.org/10.1007/978-3-642-00202-1_24","mag":"2103718624"},"language":"en","primary_location":{"is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-642-00202-1_24","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319965","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Springer Nature","Springer Science+Business Media"],"type":"book series"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false},"type":"book-chapter","type_crossref":"book-chapter","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://www.imsc.res.in/~meena/papers/kmeans.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020709378","display_name":"Meena Mahajan","orcid":"https://orcid.org/0000-0002-9116-4398"},"institutions":[{"id":"https://openalex.org/I59611059","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05078rg59","country_code":"IN","type":"facility","lineage":["https://openalex.org/I59611059"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Meena Mahajan","raw_affiliation_strings":["The Institute of Mathematical Sciences, Chennai, India 600 113#TAB#"],"affiliations":[{"raw_affiliation_string":"The Institute of Mathematical Sciences, Chennai, India 600 113#TAB#","institution_ids":["https://openalex.org/I59611059"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5042768493","display_name":"Prajakta Nimbhorkar","orcid":"https://orcid.org/0000-0002-7601-9555"},"institutions":[{"id":"https://openalex.org/I59611059","display_name":"Institute of Mathematical Sciences","ror":"https://ror.org/05078rg59","country_code":"IN","type":"facility","lineage":["https://openalex.org/I59611059"]}],"countries":["IN"],"is_corresponding":false,"raw_author_name":"Prajakta Nimbhorkar","raw_affiliation_strings":["The Institute of Mathematical Sciences, Chennai, India 600 113#TAB#"],"affiliations":[{"raw_affiliation_string":"The Institute of Mathematical Sciences, Chennai, India 600 113#TAB#","institution_ids":["https://openalex.org/I59611059"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5113712448","display_name":"Kasturi Varadarajan","orcid":null},"institutions":[{"id":"https://openalex.org/I126307644","display_name":"University of Iowa","ror":"https://ror.org/036jqmy94","country_code":"US","type":"education","lineage":["https://openalex.org/I126307644"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Kasturi Varadarajan","raw_affiliation_strings":["The University of Iowa, Iowa City, USA IA 52242-1419#TAB#"],"affiliations":[{"raw_affiliation_string":"The University of Iowa, Iowa City, USA IA 52242-1419#TAB#","institution_ids":["https://openalex.org/I126307644"]}]}],"institution_assertions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":5000,"currency":"EUR","value_usd":5392,"provenance":"doaj"},"apc_paid":null,"fwci":29.461,"has_fulltext":true,"fulltext_origin":"ngrams","cited_by_count":313,"citation_normalized_percentile":{"value":0.996528,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":null,"issue":null,"first_page":"274","last_page":"285"},"is_retracted":false,"is_paratext":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Mesh Generation Algorithms","score":0.9987,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Mesh Generation Algorithms","score":0.9987,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T11797","display_name":"Optical Code Division Multiple Access","score":0.9963,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10720","display_name":"Combinatorial Optimization and Complexity Theory","score":0.9961,"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/square","display_name":"Square (algebra)","score":0.62094265},{"id":"https://openalex.org/keywords/approximation-algorithms","display_name":"Approximation Algorithms","score":0.492441},{"id":"https://openalex.org/keywords/finite-set","display_name":"Finite set","score":0.4670234},{"id":"https://openalex.org/keywords/center","display_name":"Center (category theory)","score":0.45903066},{"id":"https://openalex.org/keywords/cutting-plane-method","display_name":"Cutting-plane method","score":0.43741488}],"concepts":[{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.62094265},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.6200839},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.6137042},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.5977613},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5956623},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5851322},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5685859},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.55436796},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5344749},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.4670234},{"id":"https://openalex.org/C2779463800","wikidata":"https://www.wikidata.org/wiki/Q5062222","display_name":"Center (category theory)","level":2,"score":0.45903066},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.45124528},{"id":"https://openalex.org/C109839438","wikidata":"https://www.wikidata.org/wiki/Q1762039","display_name":"Cutting-plane method","level":3,"score":0.43741488},{"id":"https://openalex.org/C56086750","wikidata":"https://www.wikidata.org/wiki/Q6042592","display_name":"Integer programming","level":2,"score":0.43312615},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3938799},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3444975},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27119368},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.14481795},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.14157486},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.059716225},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C121684516","wikidata":"https://www.wikidata.org/wiki/Q7600677","display_name":"Computer graphics (images)","level":1,"score":0.0},{"id":"https://openalex.org/C8010536","wikidata":"https://www.wikidata.org/wiki/Q160398","display_name":"Crystallography","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"is_oa":false,"landing_page_url":"https://doi.org/10.1007/978-3-642-00202-1_24","pdf_url":null,"source":{"id":"https://openalex.org/S106296714","display_name":"Lecture notes in computer science","issn_l":"0302-9743","issn":["0302-9743","1611-3349"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319965","https://openalex.org/P4310319900"],"host_organization_lineage_names":["Springer Nature","Springer Science+Business Media"],"type":"book series"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false},{"is_oa":true,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.331.1306","pdf_url":"http://www.imsc.res.in/~meena/papers/kmeans.pdf","source":{"id":"https://openalex.org/S4306400349","display_name":"CiteSeer X (The Pennsylvania State University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I130769515","host_organization_name":"Pennsylvania State University","host_organization_lineage":["https://openalex.org/I130769515"],"host_organization_lineage_names":["Pennsylvania State University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false}],"best_oa_location":{"is_oa":true,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.331.1306","pdf_url":"http://www.imsc.res.in/~meena/papers/kmeans.pdf","source":{"id":"https://openalex.org/S4306400349","display_name":"CiteSeer X (The Pennsylvania State University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I130769515","host_organization_name":"Pennsylvania State University","host_organization_lineage":["https://openalex.org/I130769515"],"host_organization_lineage_names":["Pennsylvania State University"],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false},"sustainable_development_goals":[],"grants":[],"datasets":[],"versions":[],"referenced_works_count":24,"referenced_works":["https://openalex.org/W1578468649","https://openalex.org/W1590224955","https://openalex.org/W1970866964","https://openalex.org/W1989650529","https://openalex.org/W1998325344","https://openalex.org/W1998739300","https://openalex.org/W2000258988","https://openalex.org/W2004791924","https://openalex.org/W2007280161","https://openalex.org/W2033387072","https://openalex.org/W2050761777","https://openalex.org/W2066577300","https://openalex.org/W2073459066","https://openalex.org/W2086959852","https://openalex.org/W2091283109","https://openalex.org/W2110105238","https://openalex.org/W2133962824","https://openalex.org/W2149977307","https://openalex.org/W2150593711","https://openalex.org/W2187617524","https://openalex.org/W2199495299","https://openalex.org/W3138330851","https://openalex.org/W4236385439","https://openalex.org/W4255110400"],"related_works":["https://openalex.org/W3126345618","https://openalex.org/W3006617646","https://openalex.org/W2110818895","https://openalex.org/W2065369945","https://openalex.org/W2048233275","https://openalex.org/W2045076853","https://openalex.org/W2038141985","https://openalex.org/W2002159391","https://openalex.org/W1991394924","https://openalex.org/W1973064890"],"abstract_inverted_index":{"In":[0],"the":[1,32,35,38,62],"k-means":[2],"problem,":[3],"we":[4,21],"are":[5],"given":[6],"a":[7],"finite":[8],"set":[9],"S":[10,45],"of":[11,34,37,41],"points":[12,26],"in":[13,44,61],"\\(\\Re^m\\),":[14],"and":[15,20],"integer":[16],"k":[17,25],"\u2265":[18],"1,":[19],"want":[22],"to":[23,30,46],"find":[24],"(centers)":[27],"so":[28],"as":[29],"minimize":[31],"sum":[33],"square":[36],"Euclidean":[39],"distance":[40],"each":[42],"point":[43],"its":[47],"nearest":[48],"center.":[49],"We":[50],"show":[51],"that":[52],"this":[53],"well-known":[54],"problem":[55],"is":[56],"NP-hard":[57],"even":[58],"for":[59],"instances":[60],"plane,":[63],"answering":[64],"an":[65],"open":[66],"question":[67],"posed":[68],"by":[69],"Dasgupta":[70],"[6].":[71]},"cited_by_api_url":"https://api.openalex.org/works?filter=cites:W2103718624","counts_by_year":[{"year":2024,"cited_by_count":4},{"year":2023,"cited_by_count":13},{"year":2022,"cited_by_count":18},{"year":2021,"cited_by_count":37},{"year":2020,"cited_by_count":48},{"year":2019,"cited_by_count":20},{"year":2018,"cited_by_count":23},{"year":2017,"cited_by_count":26},{"year":2016,"cited_by_count":14},{"year":2015,"cited_by_count":24},{"year":2014,"cited_by_count":11},{"year":2013,"cited_by_count":16},{"year":2012,"cited_by_count":25}],"updated_date":"2024-10-29T15:06:40.483300","created_date":"2016-06-24"}