AFPTAS results for common variants of bin packing: A new method to handle the small items

We consider two well-known natural variants of bin packing, and show that these packing problemsadmit asymptotic fully polynomial time approximation schemes (AFPTAS). 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 most1 (bins). It has been known for a while that the most basic problem admits an AFPTAS. In this paper,we develop methods that allow to extend this result to other variants of bin packing. Specifically, theproblems which we study in this paper, for which we design asymptotic fully polynomial time approxi-mation schemes, are the following. The first problem is Bin packing with cardinality constraints, wherea parameter k is given, such that a bin may contain up to k items. The goal is to minimize the number ofbins used. The second problem is Bin packing with rejection, where every item has a rejection penaltyassociated with it. An item needs to be either packed to a bin or rejected, and the goal is to minimize thenumber of used bins plus the total rejection penalty of unpacked items. This resolves the complexity oftwo important variants of the bin packing problem. Our approximation schemes use a novel method forpacking the small items. Specifically, we introduce the new notion of windows. A window is a space inwhich small items can be packed, and is based on the space left by large items in each configuration. Thekey point here is that the linear program does not assign small items into specific windows (located inspecific bins), but only to types of windows. This new method is the core of the improved running timesof our schemes over the running times of the previous results, which are only asymptotic polynomialtime approximation schemes (APTAS).