Analysis of Lower Bounds for Online Bin Packing with Two Item Sizes

Abstract

In the bin packing problem, we are asked to place given items, each being of size between zero and one, into bins of capacity one. The goal is to minimize the number of bins that contain at least one item. An online algorithm for the bin packing problem decides where to place each item one by one when it arrives. The asymptotic approximation ratio of the bin packing problem is defined as the performance of an optimal online algorithm for the problem. That value indicates the intrinsic hardness of the bin packing problem. In this paper we study the bin packing problem in which every item is of either size $\alpha$ or size $\beta$ $(\leq \alpha)$. While the asymptotic approximation ratio for $\alpha > \frac{1}{2}$ was already identified, that for $\alpha \leq \frac{1}{2}$ is only partially known. This paper is the first to give a lower bound on the asymptotic approximation ratio for any $\alpha \leq \frac{1}{2}$, by formulating linear optimization problems. Furthermore, we derive another lower bound in a closed form by constructing dual feasible solutions.