Improved bounds for the conflict-free chromatic art gallery problem

Abstract

In chromatic variants of the art gallery problem, simple polygons are guarded with point guards that are assigned one of k colors each. We say these guards cover the polygon. Here we consider the conflict-free chromatic art gallery problem, first studied by Bartschi and Suri (Algorithmica 2013): A covering of the polygon is conflict-free if each point of the polygon is seen by some guard whose color appears exactly once among the guards visible to that point. We are interested in the smallest number k(n) of colors that ensure such a covering for every n-vertex polygon. It is known that k(n) is O(log n) for orthogonal and for monotone polygons, and O(log2 n) for arbitrary simple polygons. Our main contribution in this paper is an improvement of the upper bound on k(n) to O(log n) for simple polygons. The bound is achieved through a partitioning of the polygon into weak visibility subpolygons, which is known as a window partition. In a weak visibility polygon, there is a boundary edge e such that each point of the polygon is seen by some point on e. We show for the first time for this special class of polygons an upper bound of O(log n). For the subpolygons of the window partition we prove a novel concept of independence that allows to reuse colors in independent subpolygons. Combining these results leads to the upper bound of O(log n) for arbitrary simple polygons.