Online Knapsack Problems with a Resource Buffer

In this paper, we introduce online knapsack problems with a resource buffer. In the problems, we are given a knapsack with capacity $1$, a buffer with capacity $R\ge 1$, and items that arrive one by one. Each arriving item has to be taken into the buffer or discarded on its arrival irrevocably. When every item has arrived, we transfer a subset of items in the current buffer into the knapsack. Our goal is to maximize the total value of the items in the knapsack. We consider four variants depending on whether items in the buffer are removable (i.e., we can remove items in the buffer) or non-removable, and proportional (i.e., the value of each item is proportional to its size) or general. For the general&non-removable case, we observe that no constant competitive algorithm exists for any $R\ge 1$. For the proportional&non-removable case, we show that a simple greedy algorithm is optimal for every $R\ge 1$. For the general&removable and the proportional&removable cases, we present optimal algorithms for small $R$ and give asymptotically nearly optimal algorithms for general $R$.