Bin Packing with Rejection Revisited

We consider the following generalization of bin packing. Each item has a size in (0,1] associated with it, as well as a rejection cost, that an algorithm must pay if it chooses not to pack this item. The cost of an algorithm is the sum of all rejection costs of rejected items plus the number of unit sized bins used for packing all other items.We first study the offline version of the problem and design an APTAS for it. This is a non-trivial generalization of the APTAS given by Fernandez de la Vega and Lueker for the standard bin packing problem. We further give an approximation algorithm of an absolute approximation ratio 3/2, where this value is best possible unless P=NP.Finally, we study an online version of the problem. For the bounded space variant, where only a constant number of bins can be open simultaneously, we design a sequence of algorithms whose competitive ratios tend to the best possible asymptotic competitive ratio. These algorithms are generalizations of bounded space algorithms for standard bin packing. We show that our algorithms have the same asymptotic competitive ratios as these known for the standard problem, for which the sequence of the competitive ratios tends to Π∞≈1.691. Furthermore, we introduce an unbounded space algorithm which achieves a much smaller asymptotic competitive ratio. All our results improve upon previous results of Dósa and He.