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. in Journal of the ACM 22(4). 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, i.e., the items are given one by one, Babel et al. provided lower bounds √ 2 ≈ 1.41421 and 1.5 on the asymptotic competitive ratio for k = 2 and 3, respectively, in Discrete Applied Mathematics 143(1-3). For k ≥ 4, some lower bounds (e.g., by van Vliet in Information Processing Letters 43(5)) 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 ≥ 4. For example, we improve 1.33333 to 1.5 for k = 4, and 1.33333 to 1.47058 for k = 5.