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

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 $k = 5$.