On the Geometric Red-Blue Set Cover Problem

Abstract

We study the variations of the geometric Red-Blue Set Cover (RBSC) problem in the plane using various geometric objects. We show that the RBSC problem with intervals on the real line is polynomial-time solvable. The problem is $\mathsf {NP}$-hard for rectangles anchored on two parallel lines and rectangles intersecting a horizontal line. The problem admits a polynomial-time algorithm for axis-parallel lines. However, if the objects are horizontal lines and vertical segments, the problem becomes $\mathsf {NP}$-hard. Further, the problem is $\mathsf {NP}$-hard for axis-parallel unit segments.

We introduce a variation of the Red-Blue Set Cover problem with the set system, the Special-Red-Blue Set Cover problem, and show that the problem is $\mathsf {APX}$-hard. We then show that several geometric variations of the problem with: (i) axis-parallel rectangles containing the origin in the plane, (ii) axis-parallel strips, (iii) axis-parallel rectangles that are intersecting exactly zero or four times, (iv) axis-parallel line segments, and (v) downward shadows of line segments, are $\mathsf {APX}$-hard by providing encodings of these problems as the Special-Red-Blue Set Cover problem. This is on the same line of the work by Chan and Grant [6], who provided the $\mathsf {APX}$-hardness results of the geometric set cover problem for the above classes of objects.