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

Document Type

Conference Article

Publication Title

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

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′⊆ 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 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 NP -hard when all the rectangles intersect a horizontal line. Finally, we prove that the problem is 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 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 APX -hard and we provide an encoding of each class of objects mentioned above as the Special-Red-Blue Hitting Set problem.

First Page

176

Last Page

191

DOI

10.1007/978-3-031-20796-9_13

Publication Date

1-1-2022

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