Exact Algorithms and Hardness Results for Geometric Red-Blue Hitting Set Problem

Abstract

We study geometric variations of the Red-Blue Hitting Set problem. Given two sets of objects R and B, colored red and blue, respectively, and a set of points P in the plane, the goal is to find a subset $$P^\prime \subseteq P$$ of points that hits all blue objects in B while hitting the minimum number of red objects in R. We study this problem for various geometric objects. We present a polynomial-time algorithm for the problem with intervals on the real line. On the other hand, we show that the problem is $$\textsf{NP}$$ -hard for axis-parallel unit segments. Next, we study the problem with axis-parallel rectangles. We give a polynomial-time algorithm for the problem when the rectangles are anchored on a horizontal line. The problem is shown to be $$\textsf{NP}$$ -hard when all the rectangles intersect a horizontal line. Finally, we prove that the problem is $$\textsf{APX}$$ -hard when the objects are axis-parallel rectangles containing the origin of the plane, axis-parallel rectangles where every two rectangles intersect exactly either zero or four times, axis-parallel line segments, axis-parallel strips, and downward shadows of segments. To achieve these $$\textsf{APX}$$ -hardness results, we first introduce a variation of the Red-Blue Hitting Set problem in a set system, called the Special-Red-Blue Hitting Set problem. We prove that the Special-Red-Blue Hitting Set problem is $$\textsf{APX}$$ -hard and we provide an encoding of each class of objects mentioned above as the Special-Red-Blue Hitting Set problem.