Computational Complexity of the Vertex-to-Point Conflict-Free Chromatic Art Gallery Problem

Abstract

The art gallery problem is to find a set of guards who together can observe every point of the interior of a polygon P. We study a chromatic variant of the problem, where each guard is assigned one of k distinct colors. A chromatic guarding is said to be conflict-free if at least one of the colors seen by every point in P is unique (i.e., each point in P is seen by some guard whose color appears exactly once among the guards visible to that point). In this paper, we consider vertex-to-point guarding, where the guards are placed on vertices of P, and they observe every point of the interior of P. The vertex-to-point conflict-free chromatic art gallery problem is to find a colored-guard set such that (i) guards are placed on P's vertices, and (ii) any point in P can see a guard of a unique color among all the visible guards. In this paper, it is shown that determining whether there exists a conflict-free chromatic vertex-guard set for a polygon with holes is NP-hard when the number of colors is k = 2.