{"id":"https://openalex.org/W3087831808","doi":"https://doi.org/10.1145/3409385","title":"A Complexity Theoretical Study of Fuzzy <i>K</i> -Means","display_name":"A Complexity Theoretical Study of Fuzzy <i>K</i> -Means","publication_year":2020,"publication_date":"2020-09-16","ids":{"openalex":"https://openalex.org/W3087831808","doi":"https://doi.org/10.1145/3409385","mag":"3087831808"},"language":"en","primary_location":{"id":"doi:10.1145/3409385","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3409385","pdf_url":null,"source":{"id":"https://openalex.org/S137348503","display_name":"ACM Transactions on Algorithms","issn_l":"1549-6325","issn":["1549-6325","1549-6333"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Algorithms","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/A5010580955","display_name":"Johannes Bl\u00f6mer","orcid":"https://orcid.org/0000-0002-6065-3535"},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Johannes Bl\u00f6mer","raw_affiliation_strings":["Paderborn University, Germany"],"affiliations":[{"raw_affiliation_string":"Paderborn University, Germany","institution_ids":["https://openalex.org/I206945453"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5088533271","display_name":"Sascha Brauer","orcid":null},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Sascha Brauer","raw_affiliation_strings":["Paderborn University, Germany"],"affiliations":[{"raw_affiliation_string":"Paderborn University, Germany","institution_ids":["https://openalex.org/I206945453"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031282818","display_name":"Kathrin Bujna","orcid":null},"institutions":[{"id":"https://openalex.org/I206945453","display_name":"Paderborn University","ror":"https://ror.org/058kzsd48","country_code":"DE","type":"education","lineage":["https://openalex.org/I206945453"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Kathrin Bujna","raw_affiliation_strings":["Paderborn University, Germany"],"affiliations":[{"raw_affiliation_string":"Paderborn University, Germany","institution_ids":["https://openalex.org/I206945453"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5010580955"],"corresponding_institution_ids":["https://openalex.org/I206945453"],"apc_list":null,"apc_paid":null,"fwci":0.1326,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.54667393,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":95},"biblio":{"volume":"16","issue":"4","first_page":"1","last_page":"25"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10637","display_name":"Advanced Clustering Algorithms Research","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11106","display_name":"Data Management and Algorithms","score":0.9947999715805054,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/T11063","display_name":"Rough Sets and Fuzzy Logic","score":0.9916999936103821,"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/generalization","display_name":"Generalization","score":0.6803352236747742},{"id":"https://openalex.org/keywords/fuzzy-logic","display_name":"Fuzzy logic","score":0.6320197582244873},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.624536395072937},{"id":"https://openalex.org/keywords/heuristics","display_name":"Heuristics","score":0.5929780602455139},{"id":"https://openalex.org/keywords/constant","display_name":"Constant (computer programming)","score":0.5419777631759644},{"id":"https://openalex.org/keywords/fuzzy-number","display_name":"Fuzzy number","score":0.5339891314506531},{"id":"https://openalex.org/keywords/mathematical-proof","display_name":"Mathematical proof","score":0.5027565956115723},{"id":"https://openalex.org/keywords/time-complexity","display_name":"Time complexity","score":0.4704139530658722},{"id":"https://openalex.org/keywords/defuzzification","display_name":"Defuzzification","score":0.46811288595199585},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.44909438490867615},{"id":"https://openalex.org/keywords/fuzzy-set-operations","display_name":"Fuzzy set operations","score":0.44877398014068604},{"id":"https://openalex.org/keywords/type-2-fuzzy-sets-and-systems","display_name":"Type-2 fuzzy sets and systems","score":0.4149669110774994},{"id":"https://openalex.org/keywords/fuzzy-classification","display_name":"Fuzzy classification","score":0.41023772954940796},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40990734100341797},{"id":"https://openalex.org/keywords/fuzzy-set","display_name":"Fuzzy set","score":0.3896486163139343},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3339686989784241},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3266485035419464},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.20535799860954285}],"concepts":[{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.6803352236747742},{"id":"https://openalex.org/C58166","wikidata":"https://www.wikidata.org/wiki/Q224821","display_name":"Fuzzy logic","level":2,"score":0.6320197582244873},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.624536395072937},{"id":"https://openalex.org/C127705205","wikidata":"https://www.wikidata.org/wiki/Q5748245","display_name":"Heuristics","level":2,"score":0.5929780602455139},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.5419777631759644},{"id":"https://openalex.org/C1883856","wikidata":"https://www.wikidata.org/wiki/Q3407463","display_name":"Fuzzy number","level":4,"score":0.5339891314506531},{"id":"https://openalex.org/C108710211","wikidata":"https://www.wikidata.org/wiki/Q11538","display_name":"Mathematical proof","level":2,"score":0.5027565956115723},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.4704139530658722},{"id":"https://openalex.org/C170260401","wikidata":"https://www.wikidata.org/wiki/Q5251735","display_name":"Defuzzification","level":5,"score":0.46811288595199585},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.44909438490867615},{"id":"https://openalex.org/C148671577","wikidata":"https://www.wikidata.org/wiki/Q5511133","display_name":"Fuzzy set operations","level":4,"score":0.44877398014068604},{"id":"https://openalex.org/C17350324","wikidata":"https://www.wikidata.org/wiki/Q7860605","display_name":"Type-2 fuzzy sets and systems","level":5,"score":0.4149669110774994},{"id":"https://openalex.org/C127385683","wikidata":"https://www.wikidata.org/wiki/Q1475696","display_name":"Fuzzy classification","level":4,"score":0.41023772954940796},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40990734100341797},{"id":"https://openalex.org/C42011625","wikidata":"https://www.wikidata.org/wiki/Q1055058","display_name":"Fuzzy set","level":3,"score":0.3896486163139343},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3339686989784241},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3266485035419464},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.20535799860954285},{"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/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","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}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1145/3409385","is_oa":false,"landing_page_url":"https://doi.org/10.1145/3409385","pdf_url":null,"source":{"id":"https://openalex.org/S137348503","display_name":"ACM Transactions on Algorithms","issn_l":"1549-6325","issn":["1549-6325","1549-6333"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319798","host_organization_name":"Association for Computing Machinery","host_organization_lineage":["https://openalex.org/P4310319798"],"host_organization_lineage_names":["Association for Computing Machinery"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"ACM Transactions on Algorithms","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7624910207","display_name":null,"funder_award_id":"BL 314/8-1","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"}],"funders":[{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W1530581016","https://openalex.org/W1977983731","https://openalex.org/W1979372206","https://openalex.org/W1981773323","https://openalex.org/W1990368529","https://openalex.org/W2023971898","https://openalex.org/W2036571822","https://openalex.org/W2039024677","https://openalex.org/W2045964207","https://openalex.org/W2071857538","https://openalex.org/W2084544490","https://openalex.org/W2091283109","https://openalex.org/W2094048240","https://openalex.org/W2109000886","https://openalex.org/W2146200992","https://openalex.org/W2150593711","https://openalex.org/W2157015572","https://openalex.org/W2161922322","https://openalex.org/W2171125141","https://openalex.org/W2229238337","https://openalex.org/W2286286503","https://openalex.org/W2584509174","https://openalex.org/W2605800822","https://openalex.org/W2979473749","https://openalex.org/W2995774564","https://openalex.org/W6944327879"],"related_works":["https://openalex.org/W2184441312","https://openalex.org/W1485268022","https://openalex.org/W2674573885","https://openalex.org/W1975005191","https://openalex.org/W2331106272","https://openalex.org/W2253053392","https://openalex.org/W1542808721","https://openalex.org/W4361984403","https://openalex.org/W3176863924","https://openalex.org/W173448675"],"abstract_inverted_index":{"The":[0],"fuzzy":[1,53,72,92],"K":[2,12,54,73,93],"-means":[3,13,55,94],"problem":[4,14,29],"is":[5,36,45],"a":[6,39,46,58],"popular":[7],"generalization":[8],"of":[9,27,42,60,68,78],"the":[10,23,28,52,66,76,88],"well-known":[11],"to":[15,86],"soft":[16],"clusterings.":[17],"In":[18],"this":[19],"article,":[20],"we":[21,63],"present":[22],"first":[24],"algorithmic":[25],"study":[26],"going":[30],"beyond":[31],"heuristics.":[32],"Our":[33],"main":[34],"result":[35],"that,":[37],"assuming":[38],"constant":[40],"number":[41],"clusters,":[43],"there":[44],"polynomial":[47],"time":[48],"approximation":[49],"scheme":[50],"for":[51,71],"problem.":[56],"As":[57],"part":[59],"our":[61,79],"analysis,":[62],"also":[64],"prove":[65],"existence":[67],"small":[69],"coresets":[70],"-means.":[74],"At":[75],"heart":[77],"proofs":[80],"are":[81],"two":[82],"novel":[83],"techniques":[84],"developed":[85],"analyze":[87],"otherwise":[89],"notoriously":[90],"difficult":[91],"objective":[95],"function.":[96]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}