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arXiv · 2411.11741

A Bicriterion Concentration Inequality and Prophet Inequalities for $k$-Fold Matroid Unions

Noga Alon·Nick Gravin·Tristan Pollner·Aviad Rubinstein·Hongao Wang·S. Matthew Weinberg·Qianfan Zhang
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Abstract

We investigate prophet inequalities with competitive ratios approaching $1$, seeking to generalize $k$-uniform matroids. We first show that large girth does not suffice: for all $k$, there exists a matroid of girth $\geq k$ and a prophet inequality instance on that matroid whose optimal competitive ratio is $\frac{1}{2}$. Next, we show $k$-fold matroid unions do suffice: we provide a prophet inequality with competitive ratio $1-O(\sqrt{\frac{\log k}{k}})$ for any $k$-fold matroid union. Our prophet inequality follows from an online contention resolution scheme. The key technical ingredient in our online contention resolution scheme is a novel bicriterion concentration inequality for arbitrary monotone $1$-Lipschitz functions over independent items which may be of independent interest. Applied to our particular setting, our bicriterion concentration inequality yields "Chernoff-strength" concentration for a $1$-Lipschitz function that is not (approximately) self-bounding.

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Noga Alon, Nick Gravin, Tristan Pollner, Aviad Rubinstein, Hongao Wang, S. Matthew Weinberg, Qianfan Zhang. 2024-11-18. A Bicriterion Concentration Inequality and Prophet Inequalities for $k$-Fold Matroid Unions. https://arxiv.org/abs/2411.11741

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