{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T07:43:03Z","timestamp":1791272583485,"version":"4.1.0"},"reference-count":30,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Comput."],"published-print":{"date-parts":[[2014,1]]},"abstract":"<jats:p>We consider the following general scheduling problem. The input consists of $n$ jobs, each with an arbitrary release time, size, and monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as total weighted flow time, total weighted tardiness, and sum of flow time squared. We give an $O(\\log \\log P )$ approximation for this problem, where $P$ is the ratio of the maximum to minimum job size. We also give an $O(1)$ approximation in the special case of identical release times. These results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions.<\/jats:p>","DOI":"10.1137\/130911317","type":"journal-article","created":{"date-parts":[[2014,9,25]],"date-time":"2014-09-25T13:06:12Z","timestamp":1411650372000},"page":"1684-1698","source":"Crossref","is-referenced-by-count":34,"title":["The Geometry of Scheduling"],"prefix":"10.1137","volume":"43","author":[{"given":"Nikhil","family":"Bansal","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kirk","family":"Pruhs","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2014,9,25]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/090779000"},{"key":"atypb2","first-page":"3","author":"Bansal N.","year":"2007","journal-title":"ACM Trans. Algorithms"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973075.125"},{"key":"atypb4","first-page":"38","author":"Bansal N.","year":"2011","journal-title":"Proceedings of APPROX-RANDOM"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/090772228"},{"key":"atypb6","first-page":"145","author":"Bansal N.","year":"2012","journal-title":"Proceedings of the European Symposium on Algorithms"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1145\/502102.502107"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/050625382"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1023\/B:JOSH.0000019681.52701.8b"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/BF02570718"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68891-4_20"},{"key":"atypb12","first-page":"106","author":"Carr R. D.","year":"2000","journal-title":"Proceedings of the ACM-SIAM Symposium on Discrete Algorithms"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"D. Chakrabarty, E. Grant, and J. Konemann,\n                      On column restricted and priority integer covering programs\n                      , in Proceedings of the Conference on Integer Programming and Combinatorial Optimization, 2010.","DOI":"10.1007\/978-3-642-13036-6_27"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973099.125"},{"key":"atypb15","first-page":"9","author":"Chekuri C.","year":"2012","journal-title":"ACM Trans. Algorithms"},{"key":"atypb16","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1145\/509907.509954","author":"Chekuri C.","year":"2002","journal-title":"Proceedings of the ACM Symposium on Theory of Computing"},{"key":"atypb17","first-page":"84","author":"Chekuri C.","year":"2001","journal-title":"Proceedings of the ACM Symposium on Theory of Computing"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2004.04.013"},{"key":"atypb19","first-page":"135","author":"Cheung M.","year":"2011","journal-title":"Proceedings of APPROX-RANDOM"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00454-006-1273-8"},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"W. H\u00f6hn, J. Mestre, and A. Wiese,\n                      How Unsplittable-Flow-Covering Helps Scheduling with Job-Dependent Cost Functions\n                      , arXiv:1403.1376 [cs.DS], 2014.","DOI":"10.1007\/978-3-662-43948-7_52"},{"key":"atypb22","first-page":"238","author":"Karakostas G.","year":"2009","journal-title":"Proceedings of the International Computing and Combinatorics Conference"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/0167-6377(82)90022-0"},{"key":"atypb24","first-page":"33","author":"Levi R.","year":"2008","journal-title":"Math. Oper. Res."},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"J. Matousek,\n                      Lectures on Discrete Geometry\n                      , Springer, New York, 2002.","DOI":"10.1007\/978-1-4613-0039-7"},{"key":"atypb26","unstructured":"J. Mestre and J. Verschae,\n                      A 4-Approximation for Scheduling on a Single Machine with General Cost Function\n                      , arXiv:1403.0298 [cs.DS], 2014."},{"key":"atypb27","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1145\/1998196.1998271","author":"Pach J.","year":"2011","journal-title":"Proceedings of the Symposium on Computational Geometry"},{"key":"atypb28","unstructured":"K. Pruhs, J. Sgall, and E. Torng,\n                      Online scheduling\n                      , in Handbook on Scheduling, CRC Press, Boca Raton, FL, 2004."},{"key":"atypb29","first-page":"11","author":"Varadarajan K. R.","year":"2009","journal-title":"Proceedings of the Symposium on Computational Geometry"},{"key":"atypb30","first-page":"641","author":"Varadarajan K. 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