Matching is one of the most fundamental and broadly applicable problems across many domains. In these diverse real-world applications, there is often a degree of uncertainty in the input which has led to the study of stochastic matching models. Here, each edge in the graph has a known, independent probability of existing derived from some prediction. Algorithms must probe edges to determine existence and match them irrevocably if they exist. Further, each vertex may have a patience constraint denoting how many of its neighboring edges can be probed. We present new ordered contention resolution schemes yielding improved approximation guarantees for some of the foundational problems studied in this area. For stochastic matching with patience constraints in general graphs, we provide a 0.382-approximate algorithm assuming each vertex has patience at least $2$. Under this assumption, we improve upon the previous best 0.31-approximation of Baveja et al. (2018). When the vertices do not have patience constraints, we describe a 0.432-approximate random order probing algorithm with several corollaries such as an improved guarantee for the Prophet Secretary problem under Edge Arrivals. Finally, for the special case of bipartite graphs with unit patience constraints on one of the partitions, we show a 0.632-approximate algorithm that improves on the recent $1/3$-guarantee of Hikima et al. (2021).

翻译：匹配是许多领域中最基础且应用最广泛的问题之一。在这些多样化的现实场景中，输入通常存在一定的不确定性，由此催生了随机匹配模型的研究。在该模型中，图中每条边根据某种预测结果存在一个已知的独立存在概率。算法需通过探测边来确认其是否存在，并在存在时不可撤销地进行匹配。此外，每个顶点可能有耐心约束，限制其可探测的邻边数量。我们提出了新型有序冲突消解方案，改进了该领域若干基础性问题的近似保证。针对一般图中带耐心约束的随机匹配，当每个顶点的耐心至少为2时，我们给出了0.382-近似算法，相比Baveja等人（2018）此前最优的0.31近似率有所提升。当顶点无耐心约束时，我们描述了一种0.432-近似随机顺序探测算法，并推导出多项推论，例如改进了边到达情形下先知秘书问题的保证。最后，针对二分图中某一分区顶点具有单位耐心约束的特殊情况，我们展示了0.632-近似算法，优于Hikima等人（2021）近期提出的1/3近似保证。