Optimal Cover of Points by Disks in a Simple Polygon

Let P be a simple polygon, and let Q be a set of points in P . We present an almostlinear time algorithm for computing a minimum cover of Q by disks that are contained in P . We then generalize the algorithm, so that it can compute a minimum cover of Q by homothets of any fixed compact convex set O of constant description complexity that are contained in P . This improves previous results of Katz and Morgenstern [19]. We also consider the minimum disk-cover problem when Q is contained in a (sufficiently narrow) annulus, and present a nearly linear algorithm for this case too.