{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T11:18:02Z","timestamp":1761218282663,"version":"3.41.0"},"reference-count":29,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100001659","name":"German Research Foundation","doi-asserted-by":"crossref","award":["BL 314\/8-1"],"award-info":[{"award-number":["BL 314\/8-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2020,10,31]]},"abstract":"<jats:p>\n            The fuzzy\n            <jats:italic>K<\/jats:italic>\n            -means problem is a popular generalization of the well-known\n            <jats:italic>K<\/jats:italic>\n            -means problem to soft clusterings. In this article, we present the first algorithmic study of the problem going beyond heuristics. Our main result is that, assuming a constant number of clusters, there is a polynomial time approximation scheme for the fuzzy\n            <jats:italic>K<\/jats:italic>\n            -means problem. As a part of our analysis, we also prove the existence of small coresets for fuzzy\n            <jats:italic>K<\/jats:italic>\n            -means. At the heart of our proofs are two novel techniques developed to analyze the otherwise notoriously difficult fuzzy\n            <jats:italic>K<\/jats:italic>\n            -means objective function.\n          <\/jats:p>","DOI":"10.1145\/3409385","type":"journal-article","created":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T16:10:34Z","timestamp":1600272634000},"page":"1-25","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["A Complexity Theoretical Study of Fuzzy\n            <i>K<\/i>\n            -Means"],"prefix":"10.1145","volume":"16","author":[{"given":"Johannes","family":"Bl\u00f6mer","sequence":"first","affiliation":[{"name":"Paderborn University, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sascha","family":"Brauer","sequence":"additional","affiliation":[{"name":"Paderborn University, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kathrin","family":"Bujna","sequence":"additional","affiliation":[{"name":"Paderborn University, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"320","published-online":{"date-parts":[[2020,9,16]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-03685-9_2"},{"volume-title":"Proceedings of the 31st International Symposium on Computational Geometry (SoCG\u201915)","author":"Awasthi P.","key":"e_1_2_1_2_1"},{"key":"e_1_2_1_3_1","doi-asserted-by":"crossref","unstructured":"C. Bajaj. 1988. The algebraic degree of geometric optimization problems. Discrete 8 Computational Geometry 3 2 (Jan. 1988) 177--191. DOI:https:\/\/doi.org\/10.1007\/BF02187906  C. Bajaj. 1988. The algebraic degree of geometric optimization problems. Discrete 8 Computational Geometry 3 2 (Jan. 1988) 177--191. DOI:https:\/\/doi.org\/10.1007\/BF02187906","DOI":"10.1007\/BF02187906"},{"volume-title":"FCM: The fuzzy -means clustering algorithm. Computers 8 Geosciences 10, 2","year":"1984","author":"Bezdek J. C.","key":"e_1_2_1_4_1"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.5555\/40899.40917"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/ICDM.2016.0094"},{"volume-title":"Proceedings of the 29th International Symposium on Algorithms and Computation (ISAAC\u201918)","author":"Bl\u00f6mer J.","key":"e_1_2_1_7_1"},{"key":"e_1_2_1_8_1","doi-asserted-by":"crossref","unstructured":"J. Bl\u00f6mer C. Lammersen M. Schmidt and C. Sohler. 2016. Theoretical analysis of the k-means algorithm\u2014A survey. In Algorithm\u00a0Engineering. Springer International 81--116. DOI:https:\/\/doi.org\/10.1007\/978-3-319-49487-6_3  J. Bl\u00f6mer C. Lammersen M. Schmidt and C. Sohler. 2016. Theoretical analysis of the k-means algorithm\u2014A survey. In Algorithm\u00a0Engineering. Springer International 81--116. DOI:https:\/\/doi.org\/10.1007\/978-3-319-49487-6_3","DOI":"10.1007\/978-3-319-49487-6_3"},{"key":"e_1_2_1_9_1","unstructured":"S. Brauer. 2019. Classification and Approximation of Geometric Location Problems. Ph.D. Dissertation. Paderborn University. DOI:https:\/\/doi.org\/10.17619\/UNIPB\/1-816  S. Brauer. 2019. Classification and Approximation of Geometric Location Problems. Ph.D. Dissertation. Paderborn University. 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