Matchings vs hitting sets among half-spaces in low dimensional euclidean spaces

TL;DR

This paper investigates the relationship between matchings and hitting sets among linearly separable sets in low-dimensional Euclidean spaces, establishing bounds in 2D and 3D, but showing limitations in higher dimensions.

Contribution

It proves that in 2D and 3D, either large matchings or small hitting sets exist for linearly separable sets, and demonstrates the failure of this property in higher dimensions.

Findings

Existence of $k$ disjoint sets or $O(k)$ hitting points in $ ext{R}^2$ and $ ext{R}^3$

Failure of the property in $ ext{R}^4$ and higher dimensions

Extension of results from pseudo-discs to linearly separable sets

Abstract

Let $\mathcal{F}$ be any collection of linearly separable sets of a set $P$ of $n$ points either in $\mathbb{R}^2$, or in $\mathbb{R}^3$. We show that for every natural number $k$ either one can find $k$ pairwise disjoint sets in $\mathcal{F}$, or there are $O(k)$ points in $P$ that together hit all sets in $\mathcal{F}$. The proof is based on showing a similar result for families $\mathcal{F}$ of sets separable by pseudo-discs in $\mathbb{R}^2$. We complement these statements by showing that analogous result fails to hold for collections of linearly separable sets in $\mathbb{R}^4$ and higher dimensional euclidean spaces.