Lattice and Non-lattice Piercing of Axis-Parallel Rectangles

TL;DR

This paper investigates the minimal density of point sets piercing all translates of axis-parallel rectangles, providing an algorithm for optimal lattice piercing and demonstrating cases where non-lattice piercing is significantly better.

Contribution

It introduces a polynomial-time algorithm for finding optimal lattice piercings for finite families of axis-parallel rectangles and shows limitations of lattice-based solutions compared to the optimal piercing.

Findings

The algorithm runs in polynomial time relative to the number of rectangles and their aspect ratios.

There exist families where lattice piercing density is significantly worse than the optimal.

Lattice piercing can be up to 20% less efficient than the optimal piercing set.

Abstract

Given a family ${\mathcal F}$ of shapes in the plane, we study what is the lowest possible density of a point set $P$ that pierces (``intersects'', ``hits'') all translates of each shape in ${\mathcal F}$. For instance, if ${\mathcal F}$ consists of two axis-parallel rectangles the best known piercing set, i.e., one with the lowest density, is a lattice. Given a finite family ${\mathcal F}$ of axis-parallel rectangles, we present an algorithm for finding an optimal ${\mathcal F}$-piercing lattice. The algorithm runs in time polynomial in the number of rectangles and the maximum aspect ratio of the rectangles in the family. No prior algorithms for this problem were known. On the other hand, we show that for every $n \geq 3$, there exists a family of $n$ axis-parallel rectangles for which the best piercing density achieved by a lattice is separated by a positive (constant) gap from the optimal piercing density for the respective family. Finally, we show that the best lattice can be sometimes worse by $20\%$ than the optimal piercing set.