Prophet Secretary and Matching: the Significance of the Largest Item

TL;DR

This paper introduces improved algorithms for the prophet secretary problem and matching, achieving higher competitive ratios than previous strategies and advancing the understanding of optimal online decision-making under uncertainty.

Contribution

It presents a new 0.688-competitive algorithm for the prophet secretary problem and a 0.641-competitive algorithm for prophet secretary matching, surpassing existing benchmarks.

Findings

Breaks the 0.675 barrier for prophet secretary algorithms

Achieves the first 0.641-competitive ratio for prophet secretary matching

Improves the state-of-the-art in weighted stochastic matching ratios

Abstract

The prophet secretary problem is a combination of the prophet inequality and the secretary problem, where elements are drawn from known independent distributions and arrive in uniformly random order. In this work, we design 1) a $0.688$-competitive algorithm, that breaks the $0.675$ barrier of blind strategies (Correa, Saona, Ziliotto, 2021), and 2) a $0.641$-competitive algorithm for the prophet secretary matching problem, that breaks the $1-1/e\approx 0.632$ barrier for the first time. Our second result also applies to the query-commit model of weighted stochastic matching and improves the state-of-the-art ratio (Derakhshan and Farhadi, 2023).