In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers $X_i$ that come from distributions $\mathcal{D}_i$, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample $Y_i$ from each distribution $\mathcal{D}_i$. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem was first studied by Correa et al. [SODA 2020] who managed to construct an algorithm that achieves a probability of $0.4529$. In this paper, we improve this probability to $0.5009$, almost matching an upper bound of $\simeq 0.5024$ which we show follows from earlier work. We also show that there is an algorithm which achieves the probability of $\simeq 0.5024$ asymptotically if no particular distribution is especially likely to yield the largest number. For the adversarial order version of the problem, we show that we can select the maximum number with a probability of $1/4$, and that this is best possible. Our work demonstrates that unlike in the case of the expected value objective studied by Rubinstein et al. [ITCS 2020], knowledge of a single sample is not enough to recover the factor of success guaranteed by full knowledge of the distribution.
翻译:在本文中, 我们调查了两个秘书问题的变体。 在这些变体中, 我们可以看到一个来自 $\ mathcal{ D ⁇ $ $x_ i$ 的序列, 这个序列来自分布的 $\ mathcal{ D ⁇ $, 这个序列是随机的或对称的。 我们不知道分配的顺序是什么, 但我们可以使用每个分配的单一样本 $_ i$ i$ $ mathcal{ D ⁇ i$ 。 在观察每个数字后, 我们不得不做出一个不可撤销的决定, 我们是否愿意接受它, 目标是尽可能地选择最大数目的概率。 这个问题的随机序列版本是由 Correa 等人 [SODO 2020] 研究的随机序列版本, 他们设法构建了一种算法, 概率为0. 452929 美元。 在本文中, 我们将这一概率提高到0. 509美元, 我们所显示的上限的上限是 。 我们所研究的最小的数值是 。