{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T07:50:08Z","timestamp":1781077808002,"version":"3.54.1"},"reference-count":23,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2011,9,1]],"date-time":"2011-09-01T00:00:00Z","timestamp":1314835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"funder":[{"DOI":"10.13039\/501100004329","name":"Javna Agencija za Raziskovalno Dejavnost RS","doi-asserted-by":"publisher","award":["J1-7218"],"award-info":[{"award-number":["J1-7218"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003549","name":"Orsz\u00e1gos Tudom\u00e1nyos Kutat\u00e1si Alapprogramok","doi-asserted-by":"publisher","award":["OTKA 67651"],"award-info":[{"award-number":["OTKA 67651"]}],"id":[{"id":"10.13039\/501100003549","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Magyary Zolt\u00e1n post-doctoral research fellowship","award":["P1-0297"],"award-info":[{"award-number":["P1-0297"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Algorithms"],"published-print":{"date-parts":[[2011,9]]},"abstract":"<jats:p>\n            We study the parameterized complexity of the\n            <jats:italic>k<\/jats:italic>\n            -center problem on a given\n            <jats:italic>n<\/jats:italic>\n            -point set\n            <jats:italic>P<\/jats:italic>\n            in \u211d\n            <jats:sup>\n              <jats:italic>d<\/jats:italic>\n            <\/jats:sup>\n            , with the dimension\n            <jats:italic>d<\/jats:italic>\n            as the parameter. We show that the rectilinear 3-center problem is fixed-parameter tractable, by giving an algorithm that runs in\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>n<\/jats:italic>\n            log\n            <jats:italic>n<\/jats:italic>\n            ) time for any fixed dimension\n            <jats:italic>d<\/jats:italic>\n            . On the other hand, we show that this is unlikely to be the case with both the Euclidean and rectilinear\n            <jats:italic>k<\/jats:italic>\n            -center problems for any\n            <jats:italic>k<\/jats:italic>\n            \u2265 2 and\n            <jats:italic>k<\/jats:italic>\n            \u2265 4 respectively. In particular, we prove that deciding whether\n            <jats:italic>P<\/jats:italic>\n            can be covered by the union of 2 balls of given radius or by the union of 4 cubes of given side length is W[1]-hard with respect to\n            <jats:italic>d<\/jats:italic>\n            , and thus not fixed-parameter tractable unless FPT=W[1]. For the Euclidean case, we also show that even an\n            <jats:italic>n<\/jats:italic>\n            <jats:sup>\n              <jats:italic>o<\/jats:italic>\n              (\n              <jats:italic>d<\/jats:italic>\n              )\n            <\/jats:sup>\n            -time algorithm does not exist, unless there is a 2\n            <jats:sup>\n              <jats:italic>o<\/jats:italic>\n              (\n              <jats:italic>n<\/jats:italic>\n              )\n            <\/jats:sup>\n            -time algorithm for\n            <jats:italic>n<\/jats:italic>\n            -variable 3SAT, that is, the Exponential Time Hypothesis fails.\n          <\/jats:p>","DOI":"10.1145\/2000807.2000811","type":"journal-article","created":{"date-parts":[[2011,9,27]],"date-time":"2011-09-27T14:02:19Z","timestamp":1317132139000},"page":"1-27","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["Geometric clustering"],"prefix":"10.1145","volume":"7","author":[{"given":"Sergio","family":"Cabello","sequence":"first","affiliation":[{"name":"University of Ljubljana and IMFM, Slovenia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Panos","family":"Giannopoulos","sequence":"additional","affiliation":[{"name":"Humboldt-Universit\u00e4t zu Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Knauer","sequence":"additional","affiliation":[{"name":"Freie Universit\u00e4t Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"D\u00e1niel","family":"Marx","sequence":"additional","affiliation":[{"name":"Budapest University of Technology and Economics, Hungary"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"G\u00fcnter","family":"Rote","sequence":"additional","affiliation":[{"name":"Freie Universit\u00e4t Berlin, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2011,9,28]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.1145\/299917.299918"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195999000170"},{"key":"e_1_2_1_3_1","volume-title":"Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 836--843","author":"Cabello S."},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/PL00009478"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(99)00019-X"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.ic.2005.05.001"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/1007352.1007391"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","DOI":"10.1145\/1077464.1077468"},{"key":"e_1_2_1_9_1","doi-asserted-by":"crossref","unstructured":"Downey R. and Fellows M. 1999. Parameterized Complexity. Springer-Verlag.  Downey R. and Fellows M. 1999. Parameterized Complexity. Springer-Verlag.","DOI":"10.1007\/978-1-4612-0515-9"},{"key":"e_1_2_1_10_1","doi-asserted-by":"crossref","unstructured":"Drezner Z. Ed. 1995. Facility Location: A Survey of Applications and Methods. Springer-Verlag.  Drezner Z. Ed. 1995. Facility Location: A Survey of Applications and Methods. Springer-Verlag.","DOI":"10.1007\/978-1-4612-5355-6"},{"key":"e_1_2_1_11_1","volume-title":"Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). 131--138","author":"Eppstein D.","year":"1997"},{"key":"e_1_2_1_12_1","volume-title":"Parameterized Complexity Theory. Texts in Theoretical Computer Science","author":"Flum J."},{"key":"e_1_2_1_13_1","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(81)90111-3"},{"key":"e_1_2_1_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/0219013"},{"key":"e_1_2_1_15_1","volume-title":"Abstracts of the 17th European Workshop on Computational Geometry (EuroCG'01)","author":"Hoffmann M.","year":"2001"},{"key":"e_1_2_1_16_1","unstructured":"Hoffmann M. 2002. A matrix search algorithm with applications to rectilinear 3-centers of polygons. Personal communication.  Hoffmann M. 2002. A matrix search algorithm with applications to rectilinear 3-centers of polygons. Personal communication."},{"key":"e_1_2_1_17_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcss.2001.1774"},{"key":"e_1_2_1_18_1","doi-asserted-by":"publisher","DOI":"10.1007\/11561071_41"},{"key":"e_1_2_1_19_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0747-7171(08)80067-3"},{"key":"e_1_2_1_20_1","doi-asserted-by":"publisher","DOI":"10.1137\/0213014"},{"key":"e_1_2_1_21_1","doi-asserted-by":"publisher","DOI":"10.1145\/258726.258828"},{"key":"e_1_2_1_22_1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195999000157"},{"key":"e_1_2_1_23_1","doi-asserted-by":"publisher","DOI":"10.1145\/237218.237255"}],"container-title":["ACM Transactions on Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2000807.2000811","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/2000807.2000811","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T11:00:03Z","timestamp":1750244403000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/2000807.2000811"}},"subtitle":["Fixed-parameter tractability and lower bounds with respect to the dimension"],"short-title":[],"issued":{"date-parts":[[2011,9]]},"references-count":23,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2011,9]]}},"alternative-id":["10.1145\/2000807.2000811"],"URL":"https:\/\/doi.org\/10.1145\/2000807.2000811","relation":{},"ISSN":["1549-6325","1549-6333"],"issn-type":[{"value":"1549-6325","type":"print"},{"value":"1549-6333","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,9]]},"assertion":[{"value":"2008-09-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2009-09-01","order":1,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2011-09-28","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}