Online Removable Knapsack Problems for Integer-Sized Items

In the online removable knapsack problem, a sequence of items, each labeled with its profit and its size, is given in one by one. At each arrival of an item, a player has to decide whether to put it into a knapsack or not. The player is also allowed to discard some of the items that are already in the knapsack. The objective is to maximize the total profit of the knapsack. Iwama and Taketomi gave an optimal algorithm for the case where the profit of each item is equal to its size. In this paper we consider a case with an additional constraint that the size of the knapsack is a positive integer N and the sizes of the items are all integral. For each of the cases N = 1, 2,..., we design an algorithm and prove its optimality. It is revealed that the competitive ratio is not monotonic with respect to N.