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Approximation schemes for ordered vector packing problems

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  • Alberto Caprara
  • Hans Kellerer
  • Ulrich Pferschy

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 ai, aj, either ai is componentwise not smaller than aj or aj is componentwise not smaller than ai. An asymptotic polynomial‐time approximation scheme is constructed 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. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003

Suggested Citation

  • Alberto Caprara & Hans Kellerer & Ulrich Pferschy, 2003. "Approximation schemes for ordered vector packing problems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(1), pages 58-69, February.
    • Handle: RePEc:wly:navres:v:50:y:2003:i:1:p:58-69
    DOI: 10.1002/nav.10058
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    Cited by:

    1. Hiroshi Fujiwara & Koji Kobayashi, 2015. "Improved lower bounds for the online bin packing problem with cardinality constraints," Journal of Combinatorial Optimization, Springer, vol. 29(1), pages 67-87, January.
    2. Otto, Alena & Li, Xiyu, 2020. "Product sequencing in multiple-piece-flow assembly lines," Omega, Elsevier, vol. 91(C).
    3. Katrin Heßler & Stefan Irnich & Tobias Kreiter & Ulrich Pferschy, 2022. "Bin packing with lexicographic objectives for loading weight- and volume-constrained trucks in a direct-shipping system," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 44(2), pages 1-43, June.
    4. Katrin Heßler & Stefan Irnich & Tobias Kreiter & Ulrich Pferschy, 2020. "Lexicographic Bin-Packing Optimization for Loading Trucks in a Direct-Shipping System," Working Papers 2009, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.

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