Online Knapsack Revisited

Abstract

We investigate the online variant of the (Multiple) Knapsack Problem: an algorithm is to pack items, of arbitrary sizes and profits, in k knapsacks (bins) without exceeding the capacity of any bin. We study two objective functions: the sum and the maximum of profits over all bins. With either objective, our problem statement captures and generalizes previously studied problems, e.g. Dual Bin Packing [1, 6] in case of the sum and Removable Knapsack [10, 11] in case of the maximum. Following previous studies, we consider two variants, depending on whether the algorithm is allowed to remove items (forever) from its bins or not, and two special cases where the profit of an item is a function of its size, in addition to the general setting. We study both deterministic and randomized algorithms; for the latter, we consider both the oblivious and the adaptive adversary model. We classify each variant as either admitting O(1)-competitive algorithms or not. We develop simple O(1)-competitive algorithms for some cases of the max-objective variant believed to be intrac because only 1-bin deterministic algorithms were considered before.