Covering and packing with homothets of limited capacity

TL;DR

This paper studies how many homothets of a convex body are needed to cover or pack a set with limited capacity, providing bounds, algorithms, and complexity results in various geometric settings.

Contribution

It establishes tight bounds for covering and packing with homothets of limited capacity, introduces generalized density concepts, and analyzes computational complexity.

Findings

The number of homothets needed is proportional to |S|/k, independent of dimension.

Existence of small weak epsilon-nets for homothets of convex bodies.

Polynomial algorithms for packing/covering with Euclidean balls.

Abstract

This work revolves around the two following questions: Given a convex body $C\subset\mathbb{R}^d$, a positive integer $k$ and a finite set $S\subset\mathbb{R}^d$ (or a finite Borel measure $\mu$ on $\mathbb{R}^d$), how many homothets of $C$ are required to cover $S$ if no homothet is allowed to cover more than $k$ points of $S$ (or have measure larger than $k$)? How many homothets of $C$ can be packed if each of them must cover at least $k$ points of $S$ (or have measure at least $k$)? We prove that, so long as $S$ is not too degenerate, the answer to both questions is $\Theta_d(\frac{|S|}{k})$, where the hidden constant is independent of $d$. This is optimal up to a multiplicative constant. Analogous results hold in the case of measures. Then we introduce a generalization of the standard covering and packing densities of a convex body $C$ to Borel measure spaces in $\mathbb{R}^d$ and, using the aforementioned bounds, we show that they are bounded from above and below, respectively, by functions of $d$. As an intermediate result, we give a simple proof the existence of weak $\epsilon$-nets of size $O(\frac{1}{\epsilon})$ for the range space induced by all homothets of $C$. Following some recent work in discrete geometry, we investigate the case $d=k=2$ in greater detail. We also provide polynomial time algorithms for constructing a packing/covering exhibiting the $\Theta_d(\frac{|S|}{k})$ bound mentioned above in the case that $C$ is an Euclidean ball. Finally, it is shown that if $C$ is a square then it is NP-hard to decide whether $S$ can be covered using $\frac{|S|}{4}$ squares containing $4$ points each.