1 research outputs found
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 is guarded by a finite set of point guards
and each guard is assigned one of colors. Such a chromatic guarding is said
to be conflict-free if each point sees at least one guard with a
unique color among all guards visible from . The goal is to establish bounds
on the function of the number of colors sufficient to guarantee
the existence of a conflict-free chromatic guarding for any -vertex polygon.
B\"artschi and Suri showed (Algorithmica, 2014) for
simple orthogonal polygons and the same bound applies to general simple
polygons (B\"artschi et al., SoCG 2014). In this paper, we assume the
r-visibility model instead of standard line visibility. Points and in
an orthogonal polygon are r-visible to each other if the rectangle spanned by
the points is contained in . For this model we show and . 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 -bound. Our results can be
interpreted as coloring results for special geometric hypergraphs.Comment: 18 pages, 9 figure