The Secretary Problem with Predicted Additive Gap

TL;DR

This paper introduces a new variant of the secretary problem where the algorithm has access to a weak prediction called the additive gap, and demonstrates that this information can significantly improve the competitive ratio beyond the classical bound.

Contribution

The paper presents the first analysis of the secretary problem with an additive gap prediction, achieving a competitive ratio of 0.4, surpassing the classical limit, and extends the approach to robustness and error range guarantees.

Findings

Achieves a competitive ratio of 0.4 with additive gap knowledge.

Surpasses the classical $1/e$ competitive ratio using weak predictions.

Provides robustness and error range guarantees for the algorithm.

Abstract

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.