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ACM"],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>\n            In this article we present a theoretical analysis of the online\n            <jats:italic>Sum-of-Squares<\/jats:italic>\n            algorithm (\n            <jats:italic>SS<\/jats:italic>\n            ) for bin packing along with several new variants.\n            <jats:italic>SS<\/jats:italic>\n            is applicable to any instance of bin packing in which the bin capacity\n            <jats:italic>B<\/jats:italic>\n            and item sizes\n            <jats:italic>s<\/jats:italic>\n            (\n            <jats:italic>a<\/jats:italic>\n            ) are integral (or can be scaled to be so), and runs in time\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>nB<\/jats:italic>\n            ). It performs remarkably well from an average case point of view: For any discrete distribution in which the optimal expected waste is sublinear,\n            <jats:italic>SS<\/jats:italic>\n            also has sublinear expected waste. For any discrete distribution where the optimal expected waste is bounded,\n            <jats:italic>SS<\/jats:italic>\n            has expected waste at most\n            <jats:italic>O<\/jats:italic>\n            (log\n            <jats:italic>n<\/jats:italic>\n            ). We also discuss several interesting variants on\n            <jats:italic>SS<\/jats:italic>\n            , including a randomized\n            <jats:italic>O<\/jats:italic>\n            (\n            <jats:italic>nB<\/jats:italic>\n            log\n            <jats:italic>B<\/jats:italic>\n            )-time online algorithm\n            <jats:italic>SS<\/jats:italic>\n            * whose expected behavior is essentially optimal for all discrete distributions. Algorithm\n            <jats:italic>SS<\/jats:italic>\n            * depends on a new linear-programming-based pseudopolynomial-time algorithm for solving the NP-hard problem of determining, given a discrete distribution\n            <jats:italic>F<\/jats:italic>\n            , just what is the growth rate for the optimal expected waste.\n          <\/jats:p>","DOI":"10.1145\/1120582.1120583","type":"journal-article","created":{"date-parts":[[2006,5,8]],"date-time":"2006-05-08T16:09:20Z","timestamp":1147104560000},"page":"1-65","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":37,"title":["On the Sum-of-Squares algorithm for bin packing"],"prefix":"10.1145","volume":"53","author":[{"given":"Janos","family":"Csirik","sequence":"first","affiliation":[{"name":"University of Szeged, Szeged, Hungary"}]},{"given":"David S.","family":"Johnson","sequence":"additional","affiliation":[{"name":"AT&amp;T Labs---Research, Florham Park, New Jersey"}]},{"given":"Claire","family":"Kenyon","sequence":"additional","affiliation":[{"name":"Brown University, Providence, Rhode Island"}]},{"given":"James B.","family":"Orlin","sequence":"additional","affiliation":[{"name":"MIT, Cambridge, Massachusetts"}]},{"given":"Peter W.","family":"Shor","sequence":"additional","affiliation":[{"name":"MIT, Cambridge, Massachusetts"}]},{"given":"Richard R.","family":"Weber","sequence":"additional","affiliation":[{"name":"University of Cambridge, Cambridge, England"}]}],"member":"320","published-online":{"date-parts":[[2006,1]]},"reference":[{"key":"e_1_2_1_1_1","volume-title":"Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM","author":"Albers S.","unstructured":"Albers , S. , and Mitzenmacher , M . 1998. Average-case analyses of first fit and random fit bin-packing . In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM , Philadelphia, PA, 290--299.]] Albers, S., and Mitzenmacher, M. 1998. Average-case analyses of first fit and random fit bin-packing. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, Philadelphia, PA, 290--299.]]"},{"key":"e_1_2_1_2_1","unstructured":"Alon N. and Spencer J. H. 1992. The Probabilistic Method. Wiley Interscience New York NY.]]  Alon N. and Spencer J. H. 1992. The Probabilistic Method. Wiley Interscience New York NY.]]"},{"key":"e_1_2_1_3_1","volume-title":"Proceedings of the 5th Workshop on Algorithm Engineering and Experimentation. SIAM","author":"Applegate D. L.","unstructured":"Applegate , D. L. , Buriol , L. , Dillard , B. , Johnson , D. S. , and Shor , P. W . 2003. The cutting-stock approach to bin-packing: Theory and experiments . In Proceedings of the 5th Workshop on Algorithm Engineering and Experimentation. SIAM , Philadelphia, PA, 1--15.]] Applegate, D. L., Buriol, L., Dillard, B., Johnson, D. S., and Shor, P. W. 2003. The cutting-stock approach to bin-packing: Theory and experiments. In Proceedings of the 5th Workshop on Algorithm Engineering and Experimentation. SIAM, Philadelphia, PA, 1--15.]]"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/103418.103446"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0895480197325936"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144501395423"},{"key":"e_1_2_1_7_1","unstructured":"Coffman Jr. E. G. Johnson D. S. McGeoch L. A. Shor P. W. and Weber R. R. 2005. Bin packing with discrete item sizes Part III: Average case behavior of FFD and BFD. In preparation.]]  Coffman Jr. E. G. Johnson D. S. McGeoch L. A. Shor P. W. and Weber R. 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SIAM, Philadelphia, PA, 557--566.]]"},{"key":"e_1_2_1_12_1","unstructured":"Csirik J. Johnson D. S. and Kenyon C. 2005. On the worst-case performance of the sum-of-squares algorithm for bin-packing. E-Print arXiv:cs.DS\/0509031 arXiv.org e-Print archive (http:\/\/arxiv.org\/archive\/cs).]]  Csirik J. Johnson D. S. and Kenyon C. 2005. On the worst-case performance of the sum-of-squares algorithm for bin-packing. E-Print arXiv:cs.DS\/0509031 arXiv.org e-Print archive (http:\/\/arxiv.org\/archive\/cs).]]"},{"key":"e_1_2_1_13_1","volume-title":"Proceedings 1999 Workshop on Algorithm Engineering and Experimentation, M. Goodrich and C. C. McGeoch, Eds. Lecture Notes in Computer Science","volume":"1619","author":"Csirik J.","unstructured":"Csirik , J. , Johnson , D. S. , Kenyon , C. , Shor , P. W. , and Weber , R. R . 1999. A self organizing bin-packing heuristic . In Proceedings 1999 Workshop on Algorithm Engineering and Experimentation, M. Goodrich and C. C. McGeoch, Eds. Lecture Notes in Computer Science , vol. 1619 , Springer-Verlag, Berlin, Germany, 246--265.]] Csirik, J., Johnson, D. S., Kenyon, C., Shor, P. W., and Weber, R. R. 1999. A self organizing bin-packing heuristic. In Proceedings 1999 Workshop on Algorithm Engineering and Experimentation, M. Goodrich and C. C. McGeoch, Eds. Lecture Notes in Computer Science, vol. 1619, Springer-Verlag, Berlin, Germany, 246--265.]]"},{"key":"e_1_2_1_14_1","unstructured":"Garey M. R. and Johnson D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-completeness. Freeman New York.]]   Garey M. R. and Johnson D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-completeness. 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