Piercing All Translates of a Set of Axis-Parallel Rectangles

TL;DR

This paper investigates the minimal density of point sets that intersect all translates of certain families of axis-parallel rectangles, providing exact solutions for two-shape families and an efficient approximation algorithm for larger families.

Contribution

It offers exact solutions for two-shape families and introduces a linear-time approximation algorithm with a ratio of 1.895 for multiple shapes.

Findings

Exact solution for two-shape families when one is more squarish

Bounds within 10% for wide and tall rectangles

Linear-time approximation algorithm with ratio 1.895 for multiple shapes

Abstract

For a given shape $S$ in the plane, one can ask what is the lowest possible density of a point set $P$ that pierces ("intersects", "hits") all translates of $S$. This is equivalent to determining the covering density of $S$ and as such is well studied. Here we study the analogous question for families of shapes where the connection to covering no longer exists. That is, we require that a single point set $P$ simultaneously pierces each translate of each shape from some family $\mathcal F$. We denote the lowest possible density of such an $\mathcal F$-piercing point set by $\pi_T(\mathcal F)$. Specifically, we focus on families $\mathcal F$ consisting of axis-parallel rectangles. When $|\mathcal F|=2$ we exactly solve the case when one rectangle is more squarish than $2\times 1$, and give bounds (within $10\,\%$ of each other) for the remaining case when one rectangle is wide and the other one is tall. When $|\mathcal F|\ge 2$ we present a linear-time constant-factor approximation algorithm for computing $\pi_T(\mathcal F)$ (with ratio $1.895$).