An improved algorithm for the submodular secretary problem with a cardinality constraint

TL;DR

This paper introduces an improved algorithm for the submodular secretary problem with a cardinality constraint, achieving a better competitive ratio than previous methods.

Contribution

The paper presents a novel algorithm that enhances the competitive ratio for the submodular secretary problem under cardinality constraints.

Findings

Achieves a competitive ratio of ((e - 1)^2) / (e^2 (1 + e))

Improves upon the best known competitive algorithms for this problem

Provides theoretical guarantees for the proposed algorithm

Abstract

We study the submodular secretary problem with a cardinality constraint. In this problem, $n$ candidates for secretaries appear sequentially in random order. At the arrival of each candidate, a decision maker must irrevocably decide whether to hire him. The decision maker aims to hire at most $k$ candidates that maximize a non-negative submodular set function. We propose an $(\mathrm{e} - 1)^2 / (\mathrm{e}^2 (1 + \mathrm{e}))$-competitive algorithm for this problem, which improves the best one known so far.