Computational Complexity of the Chromatic Dispersive Art Gallery Problem

抄録

The dispersive art gallery problem is to find a guard set for a polygon such that every pair of guards maintains the maximum possible distance from each other. In this paper, we study a chromatic variant of this problem, where each guard is assigned one of k distinct colors. The chromatic dispersive art gallery problem is to find a guard set for a polygon such that every pair of guards having the same color are placed as far apart from each other as possible. We study the decision version of this problem when the instance is a polyomino, which is the union of connected unit squares. In this paper, it is shown that determining whether there exists an r-visibility guard set for a polyomino with holes such that every two guards with the same color are placed at a distance of at least 6 is NP-complete when the number of colors is k = 2. Here, two points are r-visible if the smallest axis-aligned rectangle containing them lies entirely within the polyomino.