Stabbing and Covering Geometric Objects in the Plane
In this thesis, we consider a variety of different geometric covering and stabbing problems in the plane. Stabbing and covering geometric objects are two widely studied problems in computational geometry. These problems are closely related; there are many cases where covering problems are dual to stabbing problems.
We first study a problem that was posed by Tamir in 1987: "Given a set of geometric objects in the plane, can one decide in polynomial time whether there exists a convex polygon whose boundary stabs every object ?" This boundary is then called a convex stabber. We give an answer to this question by proving that deciding the existence of a convex stabber is NP-hard for many types of geometric objects. Additionally, we consider an optimization version and prove it to be APX-hard for most of the considered objects.
A similar problem is deciding whether geometric objects can be stabbed with the vertices of a rotated, scaled and translated copy of a given polygon. To the best of our knowledge, this problem was not studied so far and we present the first polynomial-time algorithm.
Another stabbing problem studied in this thesis, is the problem of stabbing sequences of geometric objects: Given a distance measure and two sequences of geometric objects, compute two point sequences that stab them under the condition that the distance between these point sequences is as small as possible (using the given distance measure). We state efficient algorithms for this problem where the objects are either line segments or disks and the distance measure is the discrete Fréchet distance.
Then, we consider covering problems. We study a new version of the two-center problem where the input is a set D of disks in the plane. We first study the problem of finding two smallest congruent disks such that each disk in D intersects one of these two disks. Then, we study the problem of covering the set D by two smallest congruent disks. We also investigate an optimization version. For these problems, we give efficient exact and approximation algorithms.
Finally, we investigate the problem of computing a largest area rectangle inscribed in a convex polygon on n vertices. If the order of the vertices of the polygon is given, we state approximation algorithms whose running times are only logarithmically dependent on n.