The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a $0.688$-competitive algorithm, that breaks the $0.675$ barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a $0.641$-competitive algorithm for the prophet secretary matching problem, that breaks the $1-1/e\approx 0.632$ barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).

翻译：先知秘书问题是先知不等式与秘书问题的结合，其中元素从已知的独立分布中抽取，并以均匀随机顺序到达。在本研究中，我们设计了：1）一种具有$0.688$竞争比的算法，突破了盲策略的$0.675$障碍（Correa、Saona、Ziliotto，2021年）；2）一种针对先知秘书匹配问题的$0.641$竞争比算法，首次突破了$1-1/e\approx 0.632$的障碍。我们的第二项结果同样适用于加权随机匹配的查询-提交模型，并改进了现有最佳比率（Derakhshan与Farhadi，2023年）。