We study online weighted bipartite matching of reusable resources where an adversarial sequence of requests for resources arrive over time. A resource that is matched is 'used' for a random duration, drawn independently from a resource-dependent distribution, after which it returns and is able to be matched again. We study the performance of the greedy policy, which matches requests to the resource that yields the highest reward. Previously, it was known that the greedy policy is 1/2 competitive against a clairvoyant benchmark that knows the request sequence in advance. In this work, we improve this result by introducing a parameter that quantifies the degree of reusability of the resources. Specifically, if p represents the smallest probability over the usage distributions that a matched resource returns in one time step, the greedy policy achieves a competitive ratio of $1/(2-p)$. Furthermore, when the usage distributions are geometric, we establish a stronger competitive ratio of $(1+p)/2$, which we demonstrate to be tight. Both of these results align with the known results in the two extreme scenarios: p = 0 corresponds to non-reusable resources, where 1/2 is known to be tight, while p = 1 corresponds to every resource returning immediately, where greedy is the optimal policy and hence the competitive ratio is 1. Finally, we show that both results are robust to approximations of the greedy policy. Our work demonstrates that the reusability of resources can enhance performance compared to the non-reusable setting, and that a simple greedy policy suffices when the degree of reusability is high. Our insights contribute to the understanding of how resource reusability can influence the performance of online algorithms, and highlight the potential for improved performance as the degree of reusability increases.

翻译：我们研究在线加权二分匹配问题，其中可重用资源面临随时间到达的对抗性请求序列。被匹配的资源会进入"使用"状态，持续一段随机时长（该时长独立服从资源相关的分布），随后资源回归并能够再次被匹配。我们分析了贪心策略的性能——该策略将请求分配给能带来最高收益的资源。此前已知贪心策略相对于预知请求序列的先知基准可实现1/2竞争比。本研究通过引入量化资源可重用程度的参数改进了该结果。具体而言，若p表示在资源使用后单时间步内返回的最小概率，则贪心策略可达竞争比$1/(2-p)$。此外，当使用时长服从几何分布时，我们建立了更优的竞争比$(1+p)/2$，并证明该界是紧的。这两个结果与两种极端情形下的已知结论一致：p=0对应不可重用资源（1/2被证明是紧界），而p=1对应资源即时返回（此时贪心策略为最优策略，竞争比为1）。最后，我们证明这两个结果对贪心策略的近似实现具有鲁棒性。研究表明，相较于不可重用场景，资源的可重用性能够提升算法性能，且当可重用程度较高时，简单的贪心策略即可达到最优。我们的见解有助于理解资源可重用性对在线算法性能的影响机制，并揭示随着可重用程度提升而带来的性能改进潜力。