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：我们考虑经典在线二分图匹配问题在探测-承诺模型中的求解。在该模型中，当在线顶点到达时，需基于已知边存在概率探测其邻接边是否存在。探测算法必须遵守承诺约束，即若探测到的边存在则必须用于匹配。此外，每个在线顶点具有耐心约束，限制其邻接边的探测次数。我们引入一种新的配置线性规划（LP），并证明该规划是最优离线探测算法的松弛。利用该规划，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： - 在最坏实例模型中，当顶点以均匀随机（u.a.r.）顺序到达时，可达最优$1/e$竞争比。 - 在已知独立分布（i.d.）实例模型中，当顶点以对抗顺序到达时，可达最优$1/2$竞争比；当顶点以u.a.r.顺序到达时，可达$1-1/e$竞争比。后两项结果改进了Brubach等人（Algorithmica 2020）此前在更受限的已知独立同分布（i.i.d.）实例模型下获得的$0.46$最优竞争比。我们的$1-1/e$-竞争算法与Ehsani等人（SODA 2018）针对先知秘书匹配问题取得的最佳结果相匹配。该算法具有高效性，且在已知图的特殊情况下可实现$1-1/e$近似比——这对应离线随机匹配问题。与Pollner等人（EC 2022）针对单侧耐心约束获得的$0.42$近似比相比，我们不仅改进了这一结果，还将Gamlath等人（SODA 2019）针对无界耐心约束的$1-1/e$近似比推广至更一般情形。