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



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 \(k = 5\).


Bin packing problem Online algorithm Competitive analysis Cardinality constraint 



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


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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

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