Optimal Approximations Made Easy

TL;DR

This paper provides a simple, modular proof of a key approximation result for finite set systems, making it accessible to a broader audience and educational settings.

Contribution

It offers a self-contained, intuitive proof of a fundamental approximation theorem, relying only on Chernoff's bound, suitable for teaching and wider understanding.

Findings

Simplified proof accessible to non-experts

Relies solely on Chernoff's concentration bound

Applicable in educational contexts

Abstract

The fundamental result of Li, Long, and Srinivasan on approximations of set systems has become a key tool across several communities such as learning theory, algorithms, computational geometry, combinatorics and data analysis. The goal of this paper is to give a modular, self-contained, intuitive proof of this result for finite set systems. The only ingredient we assume is the standard Chernoff's concentration bound. This makes the proof accessible to a wider audience, readers not familiar with techniques from statistical learning theory, and makes it possible to be covered in a single self-contained lecture in a geometry, algorithms or combinatorics course.