Nearly Tight Sample Complexity for Matroid Online Contention Resolution

TL;DR

This paper introduces a nearly optimal sample-based online contention resolution scheme for matroids, significantly reducing the number of samples needed to achieve competitive prophet inequalities in sequential decision problems.

Contribution

It presents a nearly tight sample complexity bound for matroid OCRS, improving previous results and nearly matching the known lower bounds, thus advancing the understanding of sample-based prophet inequalities.

Findings

Uses only $O( ext{log} ho imes ext{log}^2 ext{log} ho)$ samples, nearly matching the lower bound.

Achieves a sample-based prophet inequality with $O( ext{log} n + ext{log} ho imes ext{log}^2 ext{log} ho)$ samples.

Maintains the best known competitiveness of $rac{1}{4}- extvarepsilon$ in the adversarial setting.

Abstract

Due to their numerous applications, in particular in Mechanism Design, Prophet Inequalities have experienced a surge of interest. They describe competitive ratios for basic stopping time problems where random variables get revealed sequentially. A key drawback in the classical setting is the assumption of full distributional knowledge of the involved random variables, which is often unrealistic. A natural way to address this is via sample-based approaches, where only a limited number of samples from the distribution of each random variable is available. Recently, Fu, Lu, Gavin Tang, Wu, Wu, and Zhang (2024) showed that sample-based Online Contention Resolution Schemes (OCRS) are a powerful tool to obtain sample-based Prophet Inequalities. They presented the first sample-based OCRS for matroid constraints, which is a heavily studied constraint family in this context, as it captures many interesting settings. This allowed them to get the first sample-based Matroid Prophet Inequality, using $O(\log^4 n)$ many samples (per random variable), where $n$ is the number of random variables, while obtaining a constant competitiveness of $\frac{1}{4}-\varepsilon$. We present a nearly optimal sample-based OCRS for matroid constraints, which uses only $O(\log \rho \cdot \log^2\log\rho)$ many samples, almost matching a known lower bound of $\Omega(\log \rho)$, where $\rho \leq n$ is the rank of the matroid. Through the above-mentioned connection to Prophet Inequalities, this yields a sample-based Matroid Prophet Inequality using only $O(\log n + \log\rho \cdot \log^2\log\rho)$ many samples, and matching the competitiveness of $\frac{1}{4}-\varepsilon$, which is the best known competitiveness for the considered almighty adversary setting even when the distributions are fully known.