一类广义秘书问题的解析解
The solution of a generalized secretary problem via analytic expressions
摘要：
给定整数$1\leq k<n$，广义秘书问题的Gusein-Zade版本是选择$n$候选人中的$k$最佳者作为秘书，这些候选人以随机顺序进行面试。选择中的停止规则仅基于连续到达的相对秩。已知最佳策略可以由整数的非递减序列$(s_1, \ldots, s_k)$来描述，其中对于每个$1\leq l\leq k$具有$l\leq s_l<n$，并且相反地，任何这样的序列确定最佳策略的一般结构。我们发现了一个有限的解析表达式的成功概率时，使用的最优策略与序列$(s_1, \ldots, s_k)$。我们还研究了最优序列的构造问题，即最大化相应成功概率的序列。我们发现了有限的解析表达式，使计算的元素$s_l$的最佳序列一个接一个，从$l=k$到$l=1$。到目前为止，这些表达式是单独导出的，并且仅适用于值$k\leq 3$。
英文摘要：
Given integers $1\leq k<n$, the Gusein-Zade version of a generalized secretary problem is to choose one of the $k$ best of $n$ candidates for a secretary, which are interviewing in random order. The stopping rule in the selection is based only on the relative ranks of the successive arrivals. It is known that the best policy can be described by a non--decreasing sequence $(s_1, \ldots, s_k)$ of integers with $l\leq s_l<n$ for every $1\leq l\leq k$, and conversely, any such a sequence determines the general structure of the best policy. We found a finite analytic expression for the probability of success when using the optimal policy with a sequence $(s_1, \ldots, s_k)$. We also study the problem of the construction of the optimal sequence, i.e. a sequence which maximizes the corresponding probability of success. We discovered finite analytic expressions which enable to calculate the elements $s_l$ of an optimal sequence one by one, from $l=k$ to $l=1$. Until now, such expressions were derived separately, and only for the values $k\leq 3$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
