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arXiv · 2406.12799

Sample-Based Matroid Prophet Inequalities

Hu Fu·Pinyan Lu·Zhihao Gavin Tang·Hongxun Wu·Jinzhao Wu·Qianfan Zhang
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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.

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Hu Fu, Pinyan Lu, Zhihao Gavin Tang, Hongxun Wu, Jinzhao Wu, Qianfan Zhang. 2024-06-18. Sample-Based Matroid Prophet Inequalities. https://arxiv.org/abs/2406.12799

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