A short note about the learning-augmented secretary problem

TL;DR

This paper investigates the learning-augmented secretary problem, demonstrating that achieving both perfect consistency and optimal robustness is impossible even with constant candidate values, through a new hardness construction.

Contribution

It introduces a simple, explicit hardness construction showing the impossibility of simultaneously achieving 1-consistency and 1/e-robustness in the secretary problem with constant candidate values.

Findings

Achieving both 1-consistency and 1/e-robustness is impossible.

Hardness holds even when candidate values are constants.

Provides a new explicit hardness construction.

Abstract

We consider the secretary problem through the lens of learning-augmented algorithms. As it is known that the best possible expected competitive ratio is $1/e$ in the classic setting without predictions, a natural goal is to design algorithms that are 1-consistent and $1/e$-robust. Unfortunately, [FY24] provided hardness constructions showing that such a goal is not attainable when the candidates' true values are allowed to scale with $n$. Here, we provide a simple and explicit alternative hardness construction showing that such a goal is not achievable even when the candidates' true values are constants that do not scale with $n$.