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 F. We denote the lowest possible density of such an F-piercing point set by pi(F). Specifically, we focus on families F consisting of axis-parallel rectangles. When |F|=2 we exactly solve the case when one rectangle is more squarish than 2×1, and we give bounds (within 10 % of each other) for the remaining case when one rectangle is wide and the other one is tall. When |F|≥2 we present a linear-time constant-factor approximation algorithm for computing pi(F) (with ratio 1.895).

https://link.springer.com/chapter/10.1007/978-3-030-79987-8_21