A fast asymptotic approximation scheme for bin packing with rejection

''Bin packing with rejection'' is the following problem: Given a list of items with associated sizes and rejection costs, find a packing into unit bins of a subset of the list such that the number of bins used plus the sum of rejection costs of unpacked items is minimized. We show that bin packing with rejection can be reduced to n multiple knapsack problems and, based on techniques for the multiple knapsack problem, we give a fast asymptotic polynomial time approximation scheme,''Reject&Pack'', with time complexity O(n^O^(^@e^^^-^^^2^)). This improves a recent approximation scheme given by Epstein, which has time complexity O(n^O^(^(^@e^^^-^^^4^)^^^@e^^^^^^^-^^^^^^^1^)). We also show that Reject&Pack can be extended to variable-sized bin packing with rejection and give an asymptotic polynomial time approximation scheme.

[1]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[2]  Leen Stougie,et al.  Multiprocessor scheduling with rejection , 1996, SODA '96.

[3]  Yong He,et al.  Bin packing problems with rejection penalties and their dual problems , 2006, Inf. Comput..

[4]  G. S. Lueker,et al.  Bin packing can be solved within 1 + ε in linear time , 1981 .

[5]  Hans Kellerer,et al.  A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem , 1999, RANDOM-APPROX.

[6]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[7]  Han Hoogeveen,et al.  Preemptive scheduling with rejection , 2000, Math. Program..

[8]  Frank D. Murgolo.  An Efficient Approximation Scheme for Variable-Sized Bin Packing , 1987, SIAM J. Comput..

[9]  Oscar H. Ibarra,et al.  Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems , 1975, JACM.

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  Guochuan Zhang,et al.  On Variable-Sized Bin Packing , 2007 .

[12]  David S. Johnson,et al.  A 71/60 theorem for bin packing , 1985, J. Complex..

[13]  Eugene L. Lawler,et al.  Fast approximation algorithms for knapsack problems , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[14]  Leah Epstein.  Bin Packing with Rejection Revisited , 2008, Algorithmica.

[15]  Sanjeev Khanna,et al.  A Polynomial Time Approximation Scheme for the Multiple Knapsack Problem , 2005, SIAM J. Comput..

[16]  Deeparnab Chakrabarty,et al.  Knapsack Problems , 2008 .

[17]  Richard M. Karp,et al.  An efficient approximation scheme for the one-dimensional bin-packing problem , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[18]  Jakub Marecek,et al.  Handbook of Approximation Algorithms and Metaheuristics , 2010, Comput. J..

[19]  Edward G. Coffman,et al.  Approximation algorithms for bin packing: a survey , 1996 .