Abstract
In this paper we deal with the d-dimensional vector packing problem, which is a generalization of the classical bin packing problem in which each item has d distinct weights and each bin has d corresponding capacities. We address the case in which the vectors of weights associated with the items are totally ordered, i.e., given any two weight vectors a i, a j, either a i is componentwise not smaller than a j or a j is componentwise not smaller than a i, and construct an asymptotic polynomial-time approximation scheme for this case. As a corollary, we also obtain such a scheme for the bin packing problem with cardinality constraint, whose existence was an open question to the best of our knowledge.
We also extend the result to instances with constant Dilworth number, i.e. instances where the set of items can be partitioned into a constant number of totally ordered subsets. We use ideas from classical and recent approximation schemes for related problems, as well as a nontrivial procedure to round an LP solution associated with the packing of the small items.
The work of the first author was partially supported by CNR and MURST, Italy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Caprara, P. Toth, “Lower Bounds and Algorithms for the 2-Dimensional Vector Packing Problem”, to appear in Discrete Applied Mathematics (2001).
C. Chekuri, S. Khanna, “On Multi-dimensional Packing Problems”, in Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’99), ACM Press (1999).
W. Fernandez de la Vega, G.S. Luecker, “Bin packing can be solved within 1 + τ in linear time”, Combinatorica 1 (1981), 349–355.
M.R. Garey, R.L. Graham, D.S. Johnson, A.C. Yao, “Resource constrained scheduling as generalized bin packing”, J. Combinatorial Theory Ser. A 21 (1976), 257–298.
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988.
H. Kellerer, U. Pferschy, “Cardinality Constrained Bin-Packing Problems”, Annals of Operations Research 92 (1999), 335–348.
K.L. Krause, V.Y. Shen, H.D. Schwetman, “Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems”, Journal of the ACM 22 (1975), 522–550.
K. Maruyama, S.K. Chang, D.T. Tang, “A General Packing Algorithm for Multidimensional Resource Requirements”, International Journal Comput. Inform. Sci. 6 (1977), 131–149.
F.C.R. Spieksma, “A Branch-and-Bound Algorithm for the Two-Dimensional Vector Packing Problem”, Computers and Operations Research 21 (1994), 19–25.
G. J. Woeginger, “There is no Asymptotic PTAS for Two-Dimensional Vector Packing”, Information Processing Letters 64 (1997), 293–297.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Caprara, A., Kellerer, H., Pferschy, U. (2001). Approximation Schemes for Ordered Vector Packing Problems. In: Goemans, M., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. RANDOM APPROX 2001 2001. Lecture Notes in Computer Science, vol 2129. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44666-4_11
Download citation
DOI: https://doi.org/10.1007/3-540-44666-4_11
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42470-3
Online ISBN: 978-3-540-44666-8
eBook Packages: Springer Book Archive