Piercing All Translates of a Set of Axis-Parallel Rectangles
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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 π_T(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 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 π_T(ℱ) (with ratio 1.895).
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