On the geometric red-blue set cover problem

Abstract

Using various geometric objects, we study variations of the geometric Red-Blue Set Cover (RBSC) problem in $R^2$. Here, given two sets R and B of points, called red and blue points, respectively, and a set O of objects, the objective is to compute a subset $O^{\prime }\subseteq O$ of objects such that $O^{\prime }$ covers all the blue points in B and covers the minimum number of red points in R. We show that the RBSC problem with intervals on the real line is polynomial-time solvable. In $R^2$, the problem admits a polynomial-time algorithm for a particular case of axis-parallel lines when no two lines intersect at a red point. However, if the objects are horizontal lines and vertical segments, the problem becomes $\mathrm{NP}$-hard. It remains $\mathrm{NP}$-hard for axis-parallel unit length segments distributed arbitrarily in the plane. The problem is $\mathrm{NP}$-hard for the axis-parallel rectangles even when (i) each rectangle in O is anchored at one of the given two parallel lines, and (ii) a horizontal line intersects all the rectangles in O.

We show a variation of RBSC, called Special Red Blue Set Cover (SPECIAL-RBSC), to be $\mathrm{APX}$-hard. Next, we use this result to show $\mathrm{APX}$-hardness of the following geometric variations of RBSC problem in $R^2$ where the objects are (i) axis-parallel rectangles containing the origin, (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, by providing the encoding of these problems as the Special Red Blue Set Cover problem. These $\mathrm{APX}$-hardness results are in the same line of work by Chan and Grant (2014), who provided the $\mathrm{APX}$-hardness results for the geometric set cover problem for the above classes of objects.