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 $(\frac14 - \varepsilon)$-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.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.12799
- arXiv:
- arXiv:2406.12799
- Bibcode:
- 2024arXiv240612799F
- Keywords:
-
- Computer Science - Data Structures and Algorithms
- E-Print:
- To appear at EC'24