Bin packing problems with rejection penalties and their dual problems

In this paper we consider the following problems: we are given a set of n items {u1,....,un} and a number of unit-capacity bins. Each item ui has a size wi ∈ (0, 1] and a penalty pi ≥ 0. An item can be either rejected, in which case we pay its penalty, or put into one bin under the constraint that the total size of the items in the bin is no greater than 1. No item can be spread into more than one bin. The objective is to minimize the total number of used bins plus the total penalty paid for the rejected items. We call the problem bin packing with rejection penalties, and denote it as BPR. For the on-line BPR problem, we present an algorithm with an absolute competitive ratio of 2.618 while the lower bound is 2.343, and an algorithm with an asymptotic competitive ratio arbitrarily close to 1.75 while the lower bound is 1.540. For the off-line BPR problem, we present an algorithm with an absolute worst-case ratio of 2 while the lower bound is 1.5, and an algorithm with an asymptotic worst-case ratio of 1.5. We also study a closely related bin covering version of the problem. In this case pi means some amount of profit. If an item is rejected, we get its profit, or it can be put into a bin in such a way that the total size of the items in the bin is no smaller than 1. The objective is to maximize the number of covered bins plus the total profit of all rejected items. We call this problem bin covering with rejection (BCR). For the on-line BCR problem, we show that no algorithm can have absolute competitive ratio greater than 0, and present an algorithm with asymptotic competitive ratio 1/2, which is the best possible. For the off-line BCR problem, we also present an algorithm with an absolute worst-case ratio of 1/2 which matches the lower bound.

[1]  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).

[2]  Yong He,et al.  Bin Packing and Covering Problems with Rejection , 2005, COCOON.

[3]  Claire Mathieu,et al.  Better approximation algorithms for bin covering , 2001, SODA '01.

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

[5]  Guochuan Zhang,et al.  On-Line Scheduling of Parallel Jobs , 2004, SIROCCO.

[6]  Gerhard J. Woeginger,et al.  On-line scheduling of unit time jobs with rejection: minimizing the total completion time , 2002, Oper. Res. Lett..

[7]  Allen Van Gelder,et al.  Computer Algorithms: Introduction to Design and Analysis , 1978 .

[8]  Michael B. Richey,et al.  Improved bounds for harmonic-based bin packing algorithms , 1991, Discret. Appl. Math..

[9]  Y. He,et al.  On-Line Uniform Machine Scheduling with Rejection , 2000, Computing.

[10]  Sudipta Sengupta,et al.  Algorithms and Approximation Schemes for Minimum Lateness/Tardiness Scheduling with Rejection , 2003, WADS.

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

[12]  Steven S. Seiden.  Preemptive multiprocessor scheduling with rejection , 2001, Theor. Comput. Sci..

[13]  Y. He,et al.  Solving second order ordinary differential equations with maximal symmetry group , 1999 .

[14]  Chen Wei,et al.  TWO MATHEMATICAL MODELS AND ALGORITHMS OF INTERNET COMMUNICATIONS , 1999 .

[15]  David S. Johnson,et al.  Computers and In stractability: A Guide to the Theory of NP-Completeness. W. H Freeman, San Fran , 1979 .

[16]  Gerhard J. Woeginger,et al.  On-line Packing and Covering Problems , 1996, Online Algorithms.

[17]  D. Simchi-Levi.  New worst‐case results for the bin‐packing problem , 1994 .

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

[19]  André van Vliet,et al.  An Improved Lower Bound for On-Line Bin Packing Algorithms , 1992, Inf. Process. Lett..

[20]  Steven S. Seiden,et al.  On the online bin packing problem , 2001, JACM.

[21]  David R. Karger,et al.  Techniques for scheduling with rejection , 1998, J. Algorithms.

[22]  János Csirik,et al.  Online algorithms for a dual version of bin packing , 1988, Discret. Appl. Math..

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

[24]  Jeffrey D. Ullman,et al.  Worst-Case Performance Bounds for Simple One-Dimensional Packing Algorithms , 1974, SIAM J. Comput..

[25]  Klaus Jansen,et al.  An asymptotic fully polynomial time approximation scheme for bin covering , 2003, Theor. Comput. Sci..

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

[28]  Joseph Y.-T. Leung,et al.  On a Dual Version of the One-Dimensional Bin Packing Problem , 1984, J. Algorithms.

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