The secretary problem is one of the fundamental problems in online decision making; a tight competitive ratio for this problem of $1/\mathrm{e} \approx 0.368$ has been known since the 1960s. Much more recently, the study of algorithms with predictions was introduced: The algorithm is equipped with a (possibly erroneous) additional piece of information upfront which can be used to improve the algorithm's performance. Complementing previous work on secretary problems with prior knowledge, we tackle the following question: What is the weakest piece of information that allows us to break the $1/\mathrm{e}$ barrier? To this end, we introduce the secretary problem with predicted additive gap. As in the classical problem, weights are fixed by an adversary and elements appear in random order. In contrast to previous variants of predictions, our algorithm only has access to a much weaker piece of information: an \emph{additive gap} $c$. This gap is the difference between the highest and $k$-th highest weight in the sequence. Unlike previous pieces of advice, knowing an exact additive gap does not make the problem trivial. Our contribution is twofold. First, we show that for any index $k$ and any gap $c$, we can obtain a competitive ratio of $0.4$ when knowing the exact gap (even if we do not know $k$), hence beating the prevalent bound for the classical problem by a constant. Second, a slightly modified version of our algorithm allows to prove standard robustness-consistency properties as well as improved guarantees when knowing a range for the error of the prediction.

翻译：秘书问题是在线决策领域的基础问题之一；自20世纪60年代以来，其紧竞争比$1/\mathrm{e} \approx 0.368$已为人所知。近年来，带预测的算法研究被引入：算法在初始阶段被赋予一个（可能存在错误的）额外信息片段，可用于提升算法性能。作为对已有带先验知识秘书问题研究的补充，我们探讨以下问题：何种最弱的信息片段能让我们突破$1/\mathrm{e}$的界限？为此，我们提出了基于预测加性间隔的秘书问题。与经典问题相同，权重由对手设定，元素以随机顺序出现。与以往的预测变体不同，我们的算法仅能访问一种弱得多的信息：一个\emph{加性间隔}$c$。该间隔是序列中最高权重与第$k$高权重之间的差值。与以往的辅助信息不同，即使知道精确的加性间隔也不会使问题变得平凡。我们的贡献包括两方面。首先，我们证明对于任意索引$k$和任意间隔$c$，当知道精确间隔时（即使不知道$k$），可以获得$0.4$的竞争比，从而以常数优势超越了经典问题的普遍界限。其次，我们算法的轻微修改版本能够证明标准的鲁棒性-一致性性质，并在知道预测误差范围时提供更强的保证。