Parameterized Red-Blue Geometric Covers

Given a finite set of blue points and a finite set of red points on a plane, we define the Red-Blue Geometric Cover problem as the problem of finding the minimum number of specific geometrical objects that can cover all blue points, but none of the red points. We note that Red-Blue Geometric Cover is closely related to several optimization problems studied earlier. These include Set Cover, Dominating Set, and Unit Disk Cover. In this work, we will study the parameterized complexities of variations of Red-Blue Geometric Cover under several restrictions and parameters. We will prove that this problem, parameterized by the size of the solution and the Common Appearance Index (defined later), is fixed-parameter tractable. We also show that for restrictions, such as planarity of the graph representation of the problem or having a bounded Appearance Index (defined later), this problem is fixed-parameter tractable. However, in the general case we can only prove that this problem is in class W[2].