The chromatic art gallery problem asks for the minimum number of "colors" t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility-two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is in the worst case Θ(log log n). By contrast, the best known upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is, again in the worst case, Θ(log n). Finally, our techniques also help us establish the first non-trivial lower bound of Ω(log log n/log log log n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.
- Art gallery problem
- Hypergraph coloring
- Orthogonal polygons
Hoffmann, F., Kriegel, K., Suri, S., Verbeek, K. A. B., & Willert, M. (2018). Tight bounds for conflict-free chromatic guarding of orthogonal art galleries. Computational Geometry, 73, 24-34. https://doi.org/10.1016/j.comgeo.2018.01.003