Almost Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Galleries

We address recently proposed chromatic versions of the classic Art Gallery Problem. Assume a simple polygon P is guarded by a finite set of point guards and each guard is assigned one of t colors. Such a chromatic guarding is said to be conflict-free if each point p ∈ P sees at least one guard with a unique color among all guards visible from p. The goal is to establish bounds on the function χcf (n) of the number of colors sufficient to guarantee the existence of a conflict-free chromatic guarding for any n-vertex polygon. Bärtschi and Suri showed χcf (n) ∈ O(logn) (Algorithmica, 2014) for simple orthogonal polygons and the same bound applies to general simple polygons (Bärtschi et al., SoCG 2014). In this paper, we assume the r-visibility model instead of standard line visibility. Points p and q in an orthogonal polygon are r-visible to each other if the rectangle spanned by the points is contained in P . For this model we show χcf (n) ∈ O(log logn) and χcf (n) ∈ Ω(log logn/ log log logn). Most interestingly, we can show that the lower bound proof extends to guards with line visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau. This is the first non-trivial lower bound for this problem setting. Furthermore, for the strong chromatic version of the problem, where all guards r-visible from a point must have distinct colors, we prove a Θ(logn)-bound. Our results can be interpreted as coloring results for special geometric hypergraphs. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms, G.2.2 Graph Theory