We study the prophet secretary problem, a well-studied variant of the classic prophet inequality, where values are drawn from independent known distributions but arrive in uniformly random order. Upon seeing a value at each step, the decision-maker has to either select it and stop or irrevocably discard it. Traditionally, the chosen benchmark is the expected reward of the prophet, who knows all the values in advance and can always select the maximum one. %% In this work, we study the prophet secretary problem against a less pessimistic but equally well-motivated benchmark; the \emph{online} optimal. Here, the main goal is to find polynomial-time algorithms that guarantee near-optimal expected reward. As a warm-up, we present a quasi-polynomial time approximation scheme (QPTAS) achieving a $(1-\e)$-approximation in $O(n^{\text{poly} \log n\cdot f(\e)})$ time through careful discretization and non-trivial bundling processes. Using the toolbox developed for the QPTAS, coupled with a novel \emph{frontloading} technique that enables us to reduce the number of decisions we need to make, we are able to remove the dependence on $n$ in the exponent and obtain a polynomial time approximation scheme (PTAS) for this problem.

翻译：我们研究先知秘书问题，这是经典先知不等式的一个广受研究的变体，其中数值从独立已知分布中抽取，但以均匀随机顺序到达。在每一步看到数值后，决策者必须选择它并停止，或不可撤销地拒绝它。传统上，选定的基准是先知的期望收益，先知预先知道所有数值并总能选择最大值。本文研究先知秘书问题，其基准是一个不那么悲观但同样合理的基准：在线最优。主要目标是找到能保证接近最优期望收益的多项式时间算法。作为热身，我们通过精细离散化和非平凡捆绑过程，提出一个拟多项式时间近似方案（QPTAS），在$O(n^{\text{poly} \log n\cdot f(\e)})$时间内实现$(1-\e)$-近似。利用为QPTAS开发的工具箱，结合一种新颖的负载前置技术，该技术减少了需要做出的决策数量，我们能够消除指数中对$n$的依赖，并为此问题获得一个多项式时间近似方案（PTAS）。