Online Knapsack Problems with Limited Cuts

The (offline) maximization (resp., minimization) knapsack problem is given a set of items with weights and sizes, and the capacity of a knapsack, to maximize (resp., minimize) the total weight of selected items under the constraint that the total size of the selected items is at most (resp., at least) the capacity of the knapsack. In this paper, we study online maximization and minimization knapsack problems with limited cuts, in which 1) items are given one by one over time, i.e., after a decision is made on the current item, the next one is given, 2) items are allowed to be cut at most k (≥ 1) times, and 3) items are allowed to be removed from the knapsack. We obtain the following three results. (i) For the maximization knapsack problem, we propose a (k + 1)/kcompetitive online algorithm, and show that it is the best possible, i.e., no online algorithm can have a competitive ratio less than (k + 1)/k. (ii) For the minimization knapsack problem, we show that no online algorithm can have a constant competitive ratio. (iii) We extend the result (i) to the resource augmentation model, where an online algorithm is allowed to use a knapsack of capacity m (> 1), while the optimal algorithm uses a unit capacity knapsack.