Online Knapsack with Removal and Recourse

We analyze the proportional online knapsack problem with removal and limited recourse. The input is a sequence of item sizes; a subset of the items has to be packed into a knapsack of unit capacity such as to maximize their total size while not exceeding the knapsack capacity. In contrast to the classical online knapsack problem, packed items can be removed and a limited number of removed items can be re-inserted to the knapsack. Such re-insertion is called recourse. Without recourse, the competitive ratio is known to be approximately 1.618 (Iwama and Taketomi, ICALP 2002). We show that, even for only one use of recourse for the whole instance, the competitive ratio drops to 3/2. We prove that, with a constant number of k >= 2 uses of recourse, a competitive ratio of 1/ (root 3 - 1) <= 1.367 can be achieved and we give a lower bound of 1 + 1/(k + 1) for this case. For an extended use of recourse, i.e., allowing a constant number of k = 1 uses per step, we derive tight bounds for the competitive ratio of the problem, lying between 1 + 1/(k + 2) and 1 + 1/(k + 1). Motivated by the observation that the lower bounds heavily depend on the fact that the online algorithm does not know the end of the input sequence, we look at a scenario where an algorithm is informed when the instance ends. We show that with this information, the competitive ratio for a constant number of k >= 2 uses of recourse can be improved to strictly less than 1 + 1/(k + 1). We also show that this information improves the competitive ratio for one use of recourse per step and give a lower bound of = 1.088 and an upper bound of 4/3 in this case.