Tight lower bounds for the size of epsilon-nets

TL;DR

This paper establishes tight lower bounds on the size of epsilon-nets for certain geometric range spaces of VC-dimension 2, showing they can be as large as the known upper bounds, including for axis-parallel rectangles.

Contribution

It constructs geometric range spaces with VC-dimension 2 where epsilon-nets are provably large, matching known upper bounds and extending understanding of epsilon-net sizes in geometric settings.

Findings

Existence of geometric range spaces with epsilon-net size Omega(1/epsilon log(1/epsilon)).

Construction of range spaces with epsilon-net size Omega(1/epsilon log log(1/epsilon)).

Bounds are tight, matching known upper bounds for these classes.

Abstract

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an $\eps$-net of size $O\left(\frac{1}{\eps}\log\frac1{\eps}\right)$. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound can be attained. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest $\eps$-nets is superlinear in $\frac1{\eps}$, were found by Alon (2010). In his examples, the size of the smallest $\eps$-nets is $\Omega\left(\frac{1}{\eps}g(\frac{1}{\eps})\right)$, where $g$ is an extremely slowly growing function, closely related to the inverse Ackermann function. \smallskip We show that there exist geometrically defined range spaces, already of VC-dimension $2$, in which the size of the smallest $\eps$-nets is $\Omega\left(\frac{1}{\eps}\log\frac{1}{\eps}\right)$. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest $\eps$-nets is $\Omega\left(\frac{1}{\eps}\log\log\frac{1}{\eps}\right)$. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.