Combinatorial optimization problems in geometric settings

We consider several combinatorial optimization problems in a geometric set- ting. The first problem we consider is the problem of clustering to minimize the sum of radii. Given a positive integer k and a set of points with interpoint distances that satisfy the definition of being a "metric", we define a ball centered at some input point and having some radius as the set of all input points that are at a distance smaller than the radius of the ball from its center. We want to cover all input points using at most k balls so that the sum of the radii of the balls chosen is minimized. We show that when the points lie in some Euclidean space and the distance measure is the standard Euclidean metric, we can find an exact solution in polynomial time under standard assumptions about the model of computation.

The second problem we consider is the Network Spanner Topology Design problem. In this problem, given a set of nodes in the network, represented by points in some geometric setting - either a plane or a 1.5-D terrain, we want to compute a height assignment function h that assigns a height to a tower at every node such that the set of pairs of nodes that can form a direct link with each other under this height function forms a connected spanner. A pair of nodes can form a direct link if they are within a bounded distance B of each other and the heights of towers at the two nodes are sufficient to achieve Line-of-Sight connectivity - i.e. the straight line connecting the top of the towers lies above any obstacles. In the planar setting where the obstacles are modeled as having a certain maximum height and minimum clearance distance, we give a constant factor approximation algorithm. In the case where the points lie on a 1.5-D terrain we illustrate that it might be hard to use Computational Geometry to achieve efficient approximations.

The final problem we consider is the Multiway Barrier Cut problem. Here, given a set of points in the plane and a set of unit disk sensors also in the plane such that any path in the plane between any pair of input points hits at least one of the given sensor disks we consider the problem of finding the minimum size subset of these disks that still achieves this separation. We give a constant factor approximation algorithm for this problem.