Packing and Covering a Polygon with Geodesic Disks

Given a polygon P , for two points s and t contained in the polygon, their geodesic distance is the length of the shortest st-path within P . A geodesic disk of radius r centered at a point v ∈ P is the set of points in P whose geodesic distance to v is at most r. We present a polynomial time 2-approximation algorithm for finding a densest geodesic unit disk packing in P . Allowing arbitrary radii but constraining the number of disks to be k, we present a 4-approximation algorithm for finding a packing in P with k geodesic disks whose minimum radius is maximized. We then turn our focus on coverings of P and present a 2-approximation algorithm for covering P with k geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are NPhard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius. 1. Motivation and Related Work Packing and covering problems are among the most studied problems in discrete geometry (see [1],[4],[7], [12],[19],[20],[32],[34],[38],[39],[40],[44] for books on these topics). Nevertheless, most of the literature focus on packings and coverings using Euclidean balls, which is a somewhat unrealistic assumption for practical problems. A prominent practical example is the Facility Location (k-Center) problem (see for example [10]) in buildings or other constrained areas. In this a setting the relevant distance metric is the shortest path metric and not the Euclidean distance. Such a problem occurs when a mobile robot is navigating in a room such as Dept. of Computer Science, City University of New York, The Graduate Center, New York, NY, USA. [email protected]. Research supported by NSF grant 1017539.