An Asymptotically Optimal Algorithm for Online Stacking

Consider a storage area where arriving items are stored temporarily in bounded capacity stacks until their departure. We look into the problem of deciding where to put an arriving item with the objective of minimizing the maximum number of stacks used over time. The decision has to be made as soon as an item arrives, and we assume that we only have information on the departure times for the arriving item and the items currently at the storage area. We are only allowed to put an item on top of another item if the item below departs at a later time. We refer to this problem as online stacking. We assume that the storage time intervals are picked i.i.d. from $[0, 1] \times [0, 1]$ using an unknown distribution with a bounded probability density function. Under this mild condition, we present a simple polynomial time online algorithm and show that the competitive ratio converges to $1$ in probability. The result holds if the stack capacity is $o(\sqrt{n})$, where $n$ is the number of items, including the realistic case where the capacity is a constant. Our experiments show that our results also have practical relevance. To the best of our knowledge, we are the first to present an asymptotically optimal algorithm for online stacking, which is an important problem with many real-world applications within computational logistics.