Journal of Combinatorial Optimization

, Volume 29, Issue 1, pp 67–87 | Cite as

Improved lower bounds for the online bin packing problem with cardinality constraints

Article

Abstract

The bin packing problem has been extensively studied and numerous variants have been considered. The \(k\)-item bin packing problem is one of the variants introduced by Krause et al. (J ACM 22:522–550, 1975). In addition to the formulation of the classical bin packing problem, this problem imposes a cardinality constraint that the number of items packed into each bin must be at most \(k\). For the online setting of this problem, in which the items are given one by one, Babel et al. (Discret Appl Math 143:238–251, 2004) provided lower bounds \(\sqrt{2} \approx 1.41421\) and \(1.5\) on the asymptotic competitive ratio for \(k=2\) and \(3\), respectively. For \(k \ge 4\), some lower bounds (e.g., by van Vliet (Inf Process Lett 43:277–284, 1992) for the online bin packing problem, i.e., a problem without cardinality constraints, can be applied to this problem. In this paper we consider the online \(k\)-item bin packing problem. First, we improve the previous lower bound \(1.41421\) to \(1.42764\) for \(k=2\). Moreover, we propose a new method to derive lower bounds for general \(k\) and present improved bounds for various cases of \(k \ge 4\). For example, we improve \(1.33333\) to \(1.5\) for \(k = 4\), and \(1.33333\) to \(1.47058\) for \(k = 5\).

Keywords

Bin packing problem Online algorithm Competitive analysis Cardinality constraint 

Notes

Acknowledgments

This work was supported by KAKENHI (23700014 and 23500014).

References

  1. Balogh J, Békési J, Galambos G (2012) New lower bounds for certain classes of bin packing algorithms. Theor Comput Sci 440–441:1–13CrossRefGoogle Scholar
  2. Babel L, Chen B, Kellerer H, Kotov V (2004) Algorithms for on-line bin-packing problems with cardinality constraints. Discret Appl Math 143(1–3):238–251CrossRefMATHMathSciNetGoogle Scholar
  3. Borodin A, El-Yaniv R (1998) Online computation and competitive analysis. Cambridge University Press, CambridgeMATHGoogle Scholar
  4. Caprara A, Kellerer H, Pferschy U (2003) Approximation schemes for ordered vector packing problems. Naval Res Logist 50(1):58–69CrossRefMATHMathSciNetGoogle Scholar
  5. Epstein L, Levin A (2010) AFPTAS results for common variants of bin packing: a new method for handling the small items. SIAM J Optim 20(6):3121–3145CrossRefMATHMathSciNetGoogle Scholar
  6. Epstein L (2006) Online bin packing with cardinality constraints. SIAM J Discret Math 20(4):1015–1030CrossRefMATHGoogle Scholar
  7. Kellerer H, Pferschy U (1999) Cardinality constrained bin-packing problems. Ann Oper Res 92:335–348CrossRefMATHMathSciNetGoogle Scholar
  8. Krause KL, Shen VY, Schwetman HD (1975) Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. J ACM 22(4):522–550CrossRefMATHMathSciNetGoogle Scholar
  9. Krause KL, Shen VY, Schwetman HD (1977) Errata: “analysis of several task-scheduling algorithms for a model of multiprogramming computer systems”. J ACM 24(3):527CrossRefMathSciNetGoogle Scholar
  10. Ramanan PV, Brown DJ, Lee CC, Lee DT (1989) On-line bin packing in linear time. J Algorithms 10(3):305–326CrossRefMATHMathSciNetGoogle Scholar
  11. Seiden SS (2002) On the online bin packing problem. J ACM 49(5):640–671CrossRefMathSciNetGoogle Scholar
  12. Sleator DD, Tarjan RE (1985) Amortized efficiency of list update and paging rules. Commun ACM 28(2):202–208CrossRefMathSciNetGoogle Scholar
  13. van Vliet A (1992) An improved lower bound for on-line bin packing algorithms. Inf Process Lett 43(5):277–284CrossRefMATHGoogle Scholar
  14. Yao AC (1980) New algorithms for bin packing. J ACM 27(2):207–227CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringToyohashi University of TechnologyToyohashiJapan
  2. 2.Principles of Informatics Research DivisionNational Institute of InformaticsTokyoJapan

Personalised recommendations