Red Blue Set Cover Problem on Axis-Parallel Hyperplanes and Other Objects

TL;DR

This paper investigates the computational complexity of the Red Blue Set Cover problem, proving NP-hardness in 2D with axis-parallel lines and exploring fixed-parameter tractability and kernelization in higher dimensions and special cases.

Contribution

It establishes NP-hardness for the problem with axis-parallel lines in 2D and analyzes its fixed-parameter tractability and kernelization in higher dimensions and special geometric cases.

Findings

NP-hardness in 2D with axis-parallel lines

Fixed-parameter tractability in higher dimensions

Existence of polynomial kernels for parameterized versions

Abstract

Given a universe $\mathcal{U}=R \cup B$ of a finite set of red elements $R$, and a finite set of blue elements $B$ and a family $\mathcal{F}$ of subsets of $\mathcal{U}$, the \RBSC problem is to find a subset $\mathcal{F}'$ of $\mathcal{F}$ that covers all blue elements of $B$ and minimum number of red elements from $R$. We prove that the \RBSC problem is NP-hard even when $R$ and $B$ respectively are sets of red and blue points in ${\rm I\!R}^2$ and the sets in $\mathcal{F}$ 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 $\mathcal{U}$ is a set of points in ${\rm I\!R}^d$ and $\mathcal{F}$ is a collection of set of axis-parallel hyperplanes in ${\rm I\!R}^d$, under different parameterizations. 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 \RBSC problem for some special types of rectangles in ${\rm I\!R}^2$.