发表机构
Department of Mathematical Engineering, Universidad de Chile（智利大学数学工程系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究秘书问题中利用有限次序数查询改进决策，涵盖单选择与拟阵模型，给出各场景紧界，并证明查询可显著提升成功概率至最优。
AI 中文摘要
我们研究秘书问题，其中在线算法可以向一个可靠的神谕询问有限数量的关于未见元素的序数查询。值仅在元素到达时才被揭示，并且所有接受/拒绝决策都是不可撤销的。神谕从不揭示数值；它只回答相对于算法已观察到的信息所定义的问题。我们首先考虑单选择秘书问题。一个查询返回其值超过当前记录的未见元素的数量。我们分析了随机顺序、对抗顺序以及两种基于样本的对抗顺序，分别使用零次查询、一次查询和无限次查询。除了一个变体外，我们对所有变体都获得了紧界。在对抗顺序中，查询不会改善平凡的$1/n$保证。在随机顺序中，一次查询将最优成功概率从$1/e$提高到$0.4477\ldots$，而无限次查询达到最优值$1/2$。在基于样本的模型中，一次查询已经针对在线对抗者产生紧值$1/e$，而无限次查询针对离线对抗者给出精确值$(20\sqrt{10}-29)/81$。剩下的一次查询离线对抗情况仍未解决，我们为此给出了两个部分上界。我们还将模型扩展到拟阵秘书问题，其中我们研究了两种序数查询接口。一个简单查询识别那些单独改进观察集上当前最优值的未见元素，而一个完整查询根据相对于该最优值的权重给出未见元素的更细序数划分。在基于样本的在线对抗顺序下，一个简单查询产生一个$1/4$概率竞争的算法，而一个完整查询将其改进到最优的$1/e$。最后，在随机顺序模型中，使用无限次完整查询，我们实现了最优的$1/2$概率竞争比。
英文摘要
We study secretary problems in which the online algorithm can ask a reliable oracle a limited number of ordinal queries about unseen elements. Values are revealed only when elements arrive, and all accept/reject decisions are irrevocable. The oracle never reveals numerical values; it only answers questions defined relative to the information already observed by the algorithm. We first consider the single-choice secretary problem. A query returns the number of unseen elements whose value exceeds the current record. We analyze random order, adversarial order, and two sample-based adversarial orders, with no queries, one query, and unlimited queries. We obtain tight bounds for all variants except one. In adversarial order, queries do not improve over the trivial $1/n$ guarantee. In random order, one query improves the optimal success probability from $1/e$ to $0.4477\ldots$, and unlimited queries achieve the optimal value $1/2$. In the sample-based models, one query already yields the tight value $1/e$ against an online adversary, while unlimited queries give the exact value $(20\sqrt{10}-29)/81$ against an offline adversary. The remaining one-query offline-adversarial case is left open, and we give two partial upper bounds for it. We also extend the model to the matroid secretary problem, where we study two ordinal query interfaces. A simple query identifies the unseen elements that would individually improve the current optimum on the observed set, while a complete query gives a finer ordinal partition of the unseen elements by weight relative to that optimum. Under sample-based online adversarial order, a single simple query yields a $1/4$ probability-competitive algorithm, and a complete query improves this to the optimal $1/e$. Finally, with unlimited complete queries in the random-order model, we achieve the optimal $1/2$ probability-competitive ratio.
