Complexity of the General Chromatic Art Gallery Problem

For a polygonal region P with n vertices, a guard cover G is a set of points in P , such that any point in P can be seen from a point in G. In a colored guard cover, every element in a guard cover is assigned a color, such that no two guards with the same color have overlapping visibility regions. The Chromatic Art Gallery Problem (CAGP) asks for the minimum number of colors for which a colored guard cover exists. We provide first complexity results for the general CAGP, in which arbitrary guard positions are allowed. We show that it is already NP-hard to decide whether two colors suffice for covering a polygon with holes, as well as for any fixed number k ≥ 2 of colors. For simple polygons, we show that computing the minimum number of colors is NP-hard for Θ(n) colors.