Sample-Based Matroid Prophet Inequalities

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 $(\frac{1}{4}-\epsilon )$-competitive matroid prophet inequality with only $O_{\epsilon }(\mathrm{poly}\log n)$ samples. Our algorithm consists of two parts: (i) a novel quantile-based reduction from matroid prophet inequalities to online contention resolution schemes (OCRSs) with $O_{\epsilon }(\log n)$ samples, and (ii) a $(\frac{1}{4}-\epsilon )$-selectable matroid OCRS with $O_{\epsilon }(\mathrm{poly}\log n)$ samples which carefully addresses an adaptivity challenge.

To appear at EC'24