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Algorithmica

, Volume 62, Issue 1–2, pp 102–129 | Cite as

Approximation Schemes for Packing Splittable Items with Cardinality Constraints

  • Leah Epstein
  • Asaf Levin
  • Rob van SteeEmail author
Article

Abstract

We continue the study of bin packing with splittable items and cardinality constraints. In this problem, a set of n items must be packed into as few bins as possible. Items may be split, but each bin may contain at most k (parts of) items, where k is some given parameter. Complicating the problem further is the fact that items may be larger than 1, which is the size of a bin. The problem is known to be strongly NP-hard for any fixed value of k.

We essentially close this problem by providing an efficient polynomial-time approximation scheme (EPTAS) for most of its versions. Namely, we present an efficient polynomial time approximation scheme for k=o(n). A PTAS for k=Θ(n) does not exist unless P = NP.

Additionally, we present dual approximation schemes for k=2 and for constant values of k. Thus we show that for any ε>0, it is possible to pack the items into the optimal number of bins in polynomial time, if the algorithm may use bins of size 1+ε.

Keywords

Approximation schemes Bin packing Cardinality constraints 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HaifaHaifaIsrael
  2. 2.Faculty of Industrial Engineering and ManagementThe TechnionHaifaIsrael
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

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