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

Abstract

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 $p\in P$ sees at least one guard with a unique color among all guards visible from $p$. The goal is to establish bounds on the function $\ddot{I}{‡}_{cf}(n)$ 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 $\ddot{I}{‡}_{cf}(n)\in O(\log n)$ (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 $\ddot{I}{‡}_{cf}(n)\in O(\log \log n)$ and $\ddot{I}{‡}_{cf}(n)\in \hat{I}\text{c◯}(\log \log n/\log \log \log n)$. 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 $\hat{I}˜(\log n)$-bound. Our results can be interpreted as coloring results for special geometric hypergraphs.

18 pages, 9 figures