Hitting and Piercing Rectangles Induced by a Point Set

Abstract

We consider various hitting and piercing problems for the family of axis-parallel rectangles induced by a point set. Selection Lemmas on induced objects are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection Lemma type results typically bound the maximum number of induced objects that are hit/pierced by a single point. First, we prove an exact result on the strong and the weak variant of the First Selection Lemma for rectangles. We also show bounds for the Second Selection Lemma which improve upon previous bounds when there are near-quadratic number of induced rectangles. Next, we consider the hitting set problem for induced rectangles. This is a special case of the geometric hitting set problem which has been extensively studied. We give efficient algorithms and show exact combinatorial bounds on the hitting set problem for two special classes of induced axis-parallel rectangles. Finally, we show that the minimum hitting set problem for all induced lines is NP-Complete.