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

Abstract

Given a universe U=R∪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′ 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 IR2 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 IRd and F is a collection of set of axis−parallel hyperplanes in IRd 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 IR2.