We study the classic online bipartite matching problem with a twist: offline vertices, called resources, are $\textit{reusable}$. In particular, when a resource is matched to an online vertex it is unavailable for a deterministic time duration $d$ after which it becomes available again for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem have resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of $\textit{unit inventory}$ and give the first algorithms that are provably better than the naive greedy approach which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of $0.589$, generalizes the classic RANKING algorithm for online bipartite matching of non-reusable resources (Karp et al., 1990), by $\textit{reranking}$ resources independently over time. While reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of $0.505$, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex, and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al., 2020) to reusable resources, in order to induce negative correlation in our tie breaking step and to beat the competitive ratio of $0.5$. Both of our results also extend to the case where offline vertices have weights.

翻译：我们研究经典的在线双叶双叶配对问题与一个曲折: 离线的脊椎, 称为资源, 是$\ textit{ repreable} $。 特别是, 当资源与在线的脊椎匹配时, 在确定性的时间段里, 美元d$ 之后, 将无法使用它。 因此, 资源可以在一段时间里与许多不同的在线脊椎匹配。 尽管最近有关这个问题的工作解决了无弹性的情况, 我们每个资源都有巨大的启动状态( 即, 许多副本 ), 我们考虑( 更一般的) $\ textit{ ex state} 。 当资源与确定性( exactly) 0.5 相比, 资源在确定性贪婪时, 我们的第一个算法( 竞争比率为0. 589 ), 将典型的RANKING 算法比不可再生资源( Karp etrial ) (Karimate) (Karimate and the relifride) ral ral ral rial rial real resulate) resul laveal resulation 。 当期里, 资源比我们更像 资源比我们更经常地显示我们更稳定。