Tight bounds for conflict-free chromatic guarding of orthogonal art galleries

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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.