Online knapsack with removal and recourse

We analyze the competitive ratio of the proportional online knapsack problem with removal and limited recourse. 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. The variant with removal only was analyzed by Iwama and Taketomi (ICALP, 2002). We show that even a single use of recourse can improve the performance of an algorithm. We give lower bounds for a constant number of $k\ge 1$ uses of recourse in total, matching upper bounds for $1\le k\le 3$, and a general upper bound for any value of k. For a variant where a constant number of $k\ge 1$ uses of recourse can be used per step, we give tight bounds for all $k\ge 1$. We further look at a scenario where an algorithm is informed when the instance ends and give improved upper bounds in both variants for this case.