We study stationary online bipartite matching, where both types of nodes--offline and online--arrive according to Poisson processes. Offline nodes wait to be matched for some random time, determined by an exponential distribution, while online nodes need to be matched immediately. This model captures scenarios such as deceased organ donation and time-sensitive task assignments, where there is an inflow of patients and workers (offline nodes) with limited patience, while organs and tasks (online nodes) must be assigned upon arrival. We present an efficient online algorithm that achieves a $(1-1/e+\delta)$-approximation to the optimal online policy's reward for a constant $\delta > 0$, simplifying and improving previous work by Aouad and Sarita\c{c} (2022). Our solution combines recent online matching techniques, particularly pivotal sampling, which enables correlated rounding of tighter linear programming approximations, and a greedy-like algorithm. A key technical component is the analysis of a stochastic process that exploits subtle correlations between offline nodes, using renewal theory. A byproduct of our result is an improvement to the best-known competitive ratio--that compares an algorithm's performance to the optimal offline policy--via a $(1-1/\sqrt{e} + \eta)$-competitive algorithm for a universal constant $\eta > 0$, advancing the results of Patel and Wajc (2024).

翻译：我们研究平稳在线二分图匹配问题，其中离线节点与在线节点均按照泊松过程到达。离线节点等待匹配的时间服从指数分布，而在线节点则需立即匹配。该模型适用于诸如已故器官捐献和时间敏感任务分配等场景：患者与工作者（离线节点）以有限耐心持续到达，而器官与任务（在线节点）必须在到达时即刻分配。我们提出一种高效的在线算法，对于常数δ>0，该算法能实现(1-1/e+δ)近似比以逼近最优在线策略的收益，从而简化并改进了Aouad和Saritaç（2022）的先前工作。我们的解决方案融合了最新的在线匹配技术，特别是关键采样技术——该技术能对更紧致的线性规划近似解进行关联舍入，并结合了类贪心算法。一个关键技术组件是通过更新理论分析随机过程，以利用离线节点间微妙的关联性。本研究的副产品是通过(1-1/√e+η)竞争算法（η>0为普适常数）改进了已知最佳竞争比——该指标将算法性能与最优离线策略进行比较，从而推进了Patel和Wajc（2024）的研究成果。