Online removable knapsack with limited cuts

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 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)/k-competitive 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 in (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.