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

TL;DR

This paper develops a new bicriterion concentration inequality and applies it to improve prophet inequalities for $k$-fold matroid unions, approaching optimal competitive ratios.

Contribution

It introduces a novel bicriterion concentration inequality for monotone 1-Lipschitz functions and applies it to achieve near-optimal prophet inequalities for $k$-fold matroid unions.

Findings

Large girth does not guarantee high competitive ratios.

A prophet inequality with ratio $1 - O( oot{rac{ ext{log} k}{k}})$ is achieved for $k$-fold matroid unions.

The new concentration inequality may be of independent interest.

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.