We consider the classical online bipartite matching problem in the probe-commit model. In this problem, when an online vertex arrives, its edges must be probed to determine if they exist, based on known edge probabilities. A probing algorithm must respect commitment, meaning that if a probed edge exists, it must be used in the matching. Additionally, each online vertex has a patience constraint which limits the number of probes that can be made to an online vertex's adjacent edges. We introduce a new configuration linear program (LP) which we prove is a relaxation of an optimal offline probing algorithm. Using this LP, we establish the following competitive ratios which depend on the model used to generate the instance graph, and the arrival order of its online vertices: - In the worst-case instance model, an optimal $1/e$ ratio when the vertices arrive in uniformly at random (u.a.r.) order. - In the known independently distributed (i.d.) instance model, an optimal $1/2$ ratio when the vertices arrive in adversarial order, and a $1-1/e$ ratio when the vertices arrive in u.a.r. order. The latter two results improve upon the previous best competitive ratio of $0.46$ due to Brubach et al. (Algorithmica 2020), which only held in the more restricted known i.i.d. (independent and identically distributed) instance model. Our $1-1/e$-competitive algorithm matches the best known result for the prophet secretary matching problem due to Ehsani et al. (SODA 2018). Our algorithm is efficient and implies a $1-1/e$ approximation ratio for the special case when the graph is known. This is the offline stochastic matching problem, and we improve upon the $0.42$ approximation ratio for one-sided patience due to Pollner et al. (EC 2022), while also generalizing the $1-1/e$ approximation ratio for unbounded patience due to Gamlath et al. (SODA 2019).

翻译：我们考虑探针-提交模型中的经典在线二分图匹配问题。在该问题中，当一个在线顶点到达时，必须根据已知的边存在概率对其邻边进行探针检测以确定边是否存在。探针算法必须遵循提交约束：若某条被探针的边存在，则必须将其纳入匹配。此外，每个在线顶点具有耐心约束，限制了可对其邻边进行探针检测的次数。我们提出了一种新的配置线性规划（LP），并证明其是最优离线探针算法的松弛形式。基于该线性规划，我们建立了以下竞争比，其结果取决于实例图的生成模型及在线顶点的到达顺序：- 在最坏情况实例模型中，当顶点以均匀随机顺序到达时，获得最优的$1/e$竞争比。- 在已知独立分布实例模型中，当顶点以对抗顺序到达时获得最优的$1/2$竞争比，以均匀随机顺序到达时获得$1-1/e$竞争比。后两项结果改进了Brubach等人（Algorithmica 2020）先前在限制更强的已知独立同分布实例模型中获得的$0.46$最佳竞争比。我们$1-1/e$竞争比的算法与Ehsani等人（SODA 2018）针对先知秘书匹配问题获得的最佳已知结果相匹配。我们的算法是高效的，并为图结构已知的特殊情况推导出$1-1/e$的近似比。这对应于离线随机匹配问题，我们改进了Pollner等人（EC 2022）针对单侧耐心约束获得的$0.42$近似比，同时推广了Gamlath等人（SODA 2019）针对无界耐心约束获得的$1-1/e$近似比。