Sample-Based Matroid Prophet Inequalities

TL;DR

This paper introduces a new algorithm for matroid prophet inequalities that works with unknown distributions using only polylogarithmic samples, advancing understanding in online selection problems.

Contribution

It provides the first constant-factor competitive algorithm for general matroids with a sublinear number of samples, linking prophet inequalities to online contention resolution schemes.

Findings

Achieves a $(1/4 - ext{small } ext{epsilon})$-competitive ratio with polylogarithmic samples.

Develops a novel quantile-based reduction from prophet inequalities to OCRSs.

Designs a $(1/4 - ext{small } ext{epsilon})$-selectable matroid OCRS with efficient sampling.

Abstract

We study matroid prophet inequalities when distributions are unknown and accessible only through samples. While single-sample prophet inequalities for special matroids are known, no constant-factor competitive algorithm with even a sublinear number of samples was known for general matroids. Adding more to the stake, the single-sample version of the question for general matroids has close (two-way) connections with the long-standing matroid secretary conjecture. In this work, we give a $(\frac14 - \varepsilon)$-competitive matroid prophet inequality with only $O_\varepsilon(\mathrm{poly} \log n)$ samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with $O_\varepsilon(\log n)$ samples, and (ii) a $(\frac14 - \varepsilon)$-selectable matroid OCRS with $O_\varepsilon(\mathrm{poly} \log n)$ samples which carefully addresses an adaptivity challenge.