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A short note about the learning-augmented secretary problem

Davin Choo, Chun Kai Ling

TL;DR

This work provides a simple and explicit alternative hardness construction showing that the secretary problem goal is not achievable even when the candidates' true values are constants that do not scale with $n$.

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 $n$.

A short note about the learning-augmented secretary problem

TL;DR

This work provides a simple and explicit alternative hardness construction showing that the secretary problem goal is not achievable even when the candidates' true values are constants that do not scale with .

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 in the classic setting without predictions, a natural goal is to design algorithms that are 1-consistent and -robust. Unfortunately, [FY24] provided hardness constructions showing that such a goal is not attainable when the candidates' true values are allowed to scale with . 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 .

Paper Structure

This paper contains 3 sections, 1 theorem, 2 equations, 1 table.

Table of Contents

  1. Introduction
  2. Extension to $n > 3$.
  3. Acknowledgements.

Key Result

Theorem 1

There is a secretary instance for $n = 3$, i.e. $\{X_1, X_2, X_3\}$ will arrive in a uniform random order, whereby any 1-consistent algorithm cannot be strictly better than $1/3 + o(1)$-robust.

Theorems & Definitions (2)

  • Theorem 1
  • proof