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

TL;DR

This paper establishes tight bounds on the number of colors needed for conflict-free guarding of orthogonal polygons under r-visibility, improving previous bounds from logarithmic to double logarithmic scale.

Contribution

It introduces the first non-trivial lower bounds for conflict-free chromatic guarding under r-visibility and extends bounds to line visibility, using a novel combinatorial structure called multicolor tableau.

Findings

Conflict-free chromatic guarding bound is O(log log n) under r-visibility.

Lower bound for the problem is Omega(log log n / log log log n).

Strong chromatic guarding requires Theta(log n) colors.

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 $\chi_{cf}(n)$ of the number of colors sufficient to guarantee the existence of a conflict-free chromatic guarding for any $n$-vertex polygon. B\"artschi and Suri showed $\chi_{cf}(n)\in O(\log n)$ (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 $p$ 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 $\chi_{cf}(n)\in O(\log\log n)$ and $\chi_{cf}(n)\in \Omega(\log\log n /\log\log\log n)$. 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 $\Theta(\log n)$-bound. Our results can be interpreted as coloring results for special geometric hypergraphs.