Resource augmented semi-online bounded space bin packing

We study on-line bounded space bin-packing in the resource augmentation model of competitive analysis. In this model, the on-line bounded space packing algorithm has to pack a list LL of items with sizes in (0, 1], into a minimum number of bins of size bb, b≥1b≥1. A bounded space algorithm has the property that it only has a constant number of active bins available to accept items at any point during processing. The performance of the algorithm is measured by comparing the produced packing with an optimal offline packing of the list LL into bins of size 1. The competitive ratio then becomes a function of the on-line bin size bb. Csirik and Woeginger studied this problem in [J. Csirik, G.J. Woeginger, Resource augmentation for online bounded space bin packing, Journal of Algorithms 44(2) (2002) 308–320] and proved that no on-line bounded space algorithm can perform better than a certain bound ρ(b)ρ(b) in the worst case. We relax the on-line condition by allowing a complete repacking within the active bins, and show that the same lower bound holds for this problem as well, and repacking may only allow one to obtain the exact best possible competitive ratio of ρ(b)ρ(b) having a constant number of active bins, instead of achieving this bound in the limit. We design a polynomial time on-line algorithm that uses three active bins and achieves the exact   best possible competitive ratio ρ(b)ρ(b) for the given problem.