Red Blue Set Cover problem on axis-parallel hyperplanes and other objects

Abstract

Given a universe $U=R\cup B$ of a finite set of red elements R, and a finite set of blue elements B and a family $F$ of subsets of $U$, the Red Blue Set Cover problem is to find a subset $F^{\prime }$ of $F$ that covers all blue elements of B and minimum number of red elements from R.

We prove that the Red Blue Set Cover problem is NP-hard even when R and B respectively are sets of red and blue points in ${\mathrm{IR}}^2$ and the sets in $F$ are defined by axis−parallel lines i.e., every set is a maximal set of points with the same x or y coordinate.

We then study the parameterized complexity of a generalization of this problem, where $U$ is a set of points in ${\mathrm{IR}}^d$ and $F$ is a collection of set of axis−parallel hyperplanes in ${\mathrm{IR}}^d$ under different parameterizations, where d is a constant. For every parameter, we show that the problem is fixed-parameter tractable and also show the existence of a polynomial kernel. We further consider the Red Blue Set Cover problem for some special types of rectangles in ${\mathrm{IR}}^2$.