{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T11:14:33Z","timestamp":1760440473686},"reference-count":7,"publisher":"World Scientific Pub Co Pte Lt","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Comput. Geom. Appl."],"published-print":{"date-parts":[[2005,12]]},"abstract":"<jats:p> We study the problem of separating n points in the plane, no two of which have the same x- or y-coordinate, using a minimum number of vertical and horizontal lines avoiding the points, so that each cell of the subdivision contains at most one point. Extending previous NP-hardness results due to Freimer et al. we prove that this problem and some variants of it are APX-hard. We give a 2-approximation algorithm for this problem, and a d-approximation algorithm for the d-dimensional variant, in which the points are to be separated using axis-parallel hyperplanes. To this end, we reduce the point separation problem to the rectangle stabbing problem studied by Gaur et al. Their approximation algorithm uses LP-rounding. We present an alternative LP-rounding procedure which also works for the rectangle stabbing problem. We show that the integrality ratio of the LP is exactly 2. <\/jats:p>","DOI":"10.1142\/s0218195905001865","type":"journal-article","created":{"date-parts":[[2006,1,3]],"date-time":"2006-01-03T11:55:56Z","timestamp":1136289356000},"page":"575-590","source":"Crossref","is-referenced-by-count":16,"title":["SEPARATING POINTS BY AXIS-PARALLEL LINES"],"prefix":"10.1142","volume":"15","author":[{"given":"GRUIA","family":"C\u0102LINESCU","sequence":"first","affiliation":[{"name":"Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"ADRIAN","family":"DUMITRESCU","sequence":"additional","affiliation":[{"name":"Computer Science, University of Wisconsin\u2013Milwaukee, 3200 N. Cramer Street, Milwaukee, WI 53211, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"HOWARD","family":"KARLOFF","sequence":"additional","affiliation":[{"name":"AT&amp;T Labs\u2013Research, 180 Park Ave., Florham Park, NJ 07932, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"PENG-JUN","family":"WAN","sequence":"additional","affiliation":[{"name":"Computer Science, Illinois Institute of Technology, Chicago, IL 60616, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf6","volume-title":"Computers and Intractability: A Guide to the Theory of NP-Completeness","author":"Garey M.","year":"1979"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1006\/jagm.2002.1221"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(91)90011-K"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1016\/j.dam.2003.11.014"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/S0166-218X(00)00230-4"},{"key":"rf12","doi-asserted-by":"publisher","DOI":"10.1142\/S0218195999000157"},{"key":"rf13","volume-title":"Approximath Algorithms","author":"Vazirani V.","year":"2001"}],"container-title":["International Journal of Computational Geometry &amp; Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218195905001865","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,7]],"date-time":"2019-08-07T00:29:00Z","timestamp":1565137740000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218195905001865"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,12]]},"references-count":7,"journal-issue":{"issue":"06","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2005,12]]}},"alternative-id":["10.1142\/S0218195905001865"],"URL":"https:\/\/doi.org\/10.1142\/s0218195905001865","relation":{},"ISSN":["0218-1959","1793-6357"],"issn-type":[{"value":"0218-1959","type":"print"},{"value":"1793-6357","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,12]]}}}