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Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>We study the problem of discrete geometric packing. Here, given weighted regions (say, in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity. We provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings. Using this framework we get a flotilla of results on this problem (and also on its dual, where one wants to pick a maximum weight subset of the points when the regions have capacities). For example, for the case of fat triangles of similar size, we show an $O(1)$-approximation and prove that no PTAS is possible.<\/jats:p>","DOI":"10.1137\/120898413","type":"journal-article","created":{"date-parts":[[2017,11,14]],"date-time":"2017-11-14T13:43:59Z","timestamp":1510667039000},"page":"1745-1784","source":"Crossref","is-referenced-by-count":1,"title":["Geometric Packing under Nonuniform Constraints"],"prefix":"10.1137","volume":"46","author":[{"given":"Alina","family":"Ene","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sariel","family":"Har-Peled","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Benjamin","family":"Raichel","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,11,14]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/S0925-7721(98)00028-5"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1137\/120891241"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/090762968"},{"key":"atypb4","unstructured":"P. Cappanera,\n                      A Survey on Obnoxious Facility Location Problems\n                      , Technical report, University of Pisa, 1999."},{"key":"atypb5","doi-asserted-by":"crossref","unstructured":"P. Chalermsook and J. Chuzhoy,\n                      Maximum independent set of rectangles\n                      , in Proceedings of the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2009, pp. 892-901.","DOI":"10.1137\/1.9781611973068.97"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/S0196-6774(02)00294-8"},{"key":"atypb7","first-page":"293","author":"Chan T. M.","year":"2012","journal-title":"New York"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"T. M. Chan and S. Har-Peled,\n                      Approximation algorithms for maximum independent set of pseudo-disks\n                      , in Proceedings of the 25th Annual ACM Symposium on Computational Geometry, 2009, pp. 333-340; also available online fromhttp:\/\/www.cs.uiuc.edu\/ sariel\/papers\/08\/w_indep.","DOI":"10.1145\/1542362.1542420"},{"key":"atypb9","unstructured":"T. M. Chan and S. Har-Peled,\n                      Approximation Algorithms for Maximum Independent Set of Pseudo-Disks\n                      , CoRR, abs\/1103.1431, 2011."},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"C. Chekuri, K. L. Clarkson, and S. Har-Peled,\n                      On the set multi-cover problem in geometric settings\n                      , in Proceedings of the 25th Annual ACM Symposium on Computational Geometry, 2009, pp. 341-350.","DOI":"10.1145\/1542362.1542421"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"C. Chekuri, J. Vondr\u00e1k, and R. Zenklusen,\n                      Submodular Function Maximization via the Multilinear Relaxation and Contention Resolution Schemes\n                      , CoRR, abs\/1105.4593, 2011.","DOI":"10.1145\/1993636.1993740"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/050629276"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"K. L. Clarkson,\n                      Applications of random sampling in computational geometry\n                      , II, in Proceedngs of the 4th Annual ACM Symposium on Computational Geometry, 1988, pp. 1-11.","DOI":"10.1145\/73393.73394"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02187740"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"M. de Berg, O. Cheong, M. van Kreveld, and M. H. Overmars,\n                      Computational Geometry: Algorithms and Applications\n                      , 3rd ed., Springer-Verlag, New York, 2008; also available online fromhttp:\/\/www.cs.uu.nl\/geobook\/.","DOI":"10.1007\/978-3-540-77974-2"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539702402676"},{"key":"atypb17","first-page":"233","author":"Ezra E.","year":"2013","journal-title":"New York"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"E. Ezra, B. Aronov, and M. Sharir,\n                      Improved bound for the union of fat triangles\n                      , in Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011, pp. 1778-1785.","DOI":"10.1137\/1.9781611973082.136"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"J. Fox and J. Pach,\n                      Computing the independence number of intersection graphs\n                      , in Proceedings of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), 2011, pp. 1161-1165.","DOI":"10.1137\/1.9781611973082.87"},{"key":"atypb20","unstructured":"E. Grant and T. M. Chan,\n                      Exact algorithms and APX-hardness results for geometric set cover\n                      , in Proceedings of the 23rd Canadian Conference on Computational Geometry, 2011."},{"key":"atypb21","unstructured":"S. Har-Peled,\n                      Being Fat and Friendly Is Not Enough\n                      , CoRR, abs\/0908.2369, 2009."},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"S. Har-Peled,\n                      Geometric Approximation Algorithms\n                      , AMS, Providence, RI, 2011.","DOI":"10.1090\/surv\/173"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-010-9248-1"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392825"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/0020-0190(91)90246-E"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/S009753979018330X"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"R. Motwani and P. Raghavan,\n                      Randomized Algorithms\n                      , Cambridge University Press, Cambridge, 1995.","DOI":"10.1017\/CBO9780511814075"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-010-9285-9"},{"key":"atypb29","unstructured":"A. Srinivasan,\n                      New approaches to covering and packing problems\n                      , in Proceedings of the 12th ACM-SIAM Discrete Algorithms Symposium (SODA), 2001, pp. 567-576."},{"key":"atypb30","unstructured":"P. G. Szab\u00f3, M. C. Mark\u00f3t, T. Csendes, E. Specht, L. G. Casado, and I. Garca\u0342a,\n                      New Approaches to Circle Packing in a Square: With Program Codes\n                      , Springer Optim. Appl., Springer-Verlag New York, 2007."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/0404048"},{"key":"atypb32","doi-asserted-by":"crossref","unstructured":"K. Varadarajan,\n                      Weighted geometric set cover via quasi-uniform sampling\n                      , in Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, 2010, pp. 641-648.","DOI":"10.1145\/1806689.1806777"},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"D. P. Williamson and D. B. Shmoys,\n                      The Design of Approximation Algorithms\n                      , Cambridge University Press, Cambridge, 2011.","DOI":"10.1017\/CBO9780511921735"}],"container-title":["SIAM Journal on Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/120898413","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:11:15Z","timestamp":1787339475000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/120898413"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":33,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/120898413"],"URL":"https:\/\/doi.org\/10.1137\/120898413","relation":{},"ISSN":["0097-5397","1095-7111"],"issn-type":[{"value":"0097-5397","type":"print"},{"value":"1095-7111","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}