AFPTAS Results for Common Variants of Bin Packing: A New Method for Handling the Small Items

We consider two well-known natural variants of bin packing and show that these packing problems admit asymptotic fully polynomial time approximation schemes (AFPTASs). In bin packing problems, a set of one-dimensional items of size at most 1 is to be assigned (packed) to subsets of sum at most 1 (bins). It has been known for a while that the most basic problem admits an AFPTAS. In this paper, we develop new methods that allow us to extend this result to other variants of bin packing consisting of a family of two-parameter bin packing problems. We demonstrate the use of our methods by designing AFPTASs for the following problems. The first problem is bin packing with cardinality constraints, where a parameter $k$ is given such that a bin may contain up to $k$ items. The goal is to minimize the number of bins used. The second problem is bin packing with rejection, where every item has a rejection penalty associated with it. An item needs to be either packed to a bin or rejected, and the goal is to minimize the number of bins used and the total rejection penalty of unpacked items. This resolves the complexity of two important variants of the bin packing problem. Our approximation schemes use a novel method for packing the small items. This new method is the core of the improved running times of our schemes over the running times of the previous results, which are only asymptotic polynomial time approximationschemes.