In the secretary problem we are faced with an online sequence of elements with values. Upon seeing an element we have to make an irrevocable take-it-or-leave-it decision. The goal is to maximize the probability of picking the element of maximum value. The most classic version of the problem is that in which the elements arrive in random order and their values are arbitrary. However, by varying the available information, new interesting problems arise. Also the case in which the arrival order is adversarial instead of random leads to interesting variants that have been considered in the literature. In this paper we study both the random order and adversarial order secretary problems with an additional twist. The values are arbitrary, but before starting the online sequence we independently sample each element with a fixed probability $p$. The sampled elements become our information or history set and the game is played over the remaining elements. We call these problems the random order secretary problem with $p$-sampling (ROS$p$ for short) and the adversarial order secretary problem with $p$-sampling (AOS$p$ for short). Our main result is to obtain best possible algorithms for both problems and all values of $p$. As $p$ grows to 1 the obtained guarantees converge to the optimal guarantees in the full information case. In the adversarial order setting, the best possible algorithm turns out to be a simple fixed threshold algorithm in which the optimal threshold is a function of $p$ only. In the random order setting we prove that the best possible algorithm is characterized by a fixed sequence of time thresholds, dictating at which point in time we should start accepting a value that is both a maximum of the online sequence and has a given ranking within the sampled elements.

中文翻译：

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独立抽样的秘书问题

在秘书问题中，我们面临一个带有值的在线元素序列。在看到一个元素时，我们必须做出不可撤销的接受或放弃的决定。目标是最大化选择最大值元素的概率。该问题最经典的版本是元素以随机顺序到达并且它们的值是任意的。然而，通过改变可用信息，新的有趣问题出现了。此外，到达顺序是对抗性而不是随机的情况会导致文献中考虑过的有趣变体。在本文中，我们研究了随机顺序和对抗性顺序秘书问题，并进行了额外的研究。这些值是任意的，但在开始在线序列之前，我们以固定概率 $p$ 独立采样每个元素。采样的元素成为我们的信息或历史集，游戏在剩余的元素上进行。我们将这些问题称为带有 $p$-sampling（简称 ROS$p$）的随机订单秘书问题和带有 $p$-sampling（简称 AOS$p$）的对抗订单秘书问题。我们的主要结果是为两个问题和 $p$ 的所有值获得最佳算法。随着 $p$ 增长到 1，获得的保证收敛到全信息情况下的最佳保证。在对抗性顺序设置中，最好的算法被证明是一个简单的固定阈值算法，其中最佳阈值仅是 $p$ 的函数。在随机顺序设置中，我们证明了最好的算法的特点是固定的时间阈值序列，