In this paper we present some new, practical, geometric optimization techniques for computing polygon partitions, 1D and 2D point, interval, square and rectangle covers, as well as 1D and 2D interval and rectangle K-centers. All the techniques we present have immediate applications to several cost optimization and facility location problems which are quite common in practice. The main technique...

We give an algorithm for determining an optimal step function approximation of weighted data, where the error is measured with respect to the L∞ norm. The algorithm takes Θ(n+ log n · b(1 + log n/b)) time and Θ(n) space, where b is the number of steps. Thus the time is Θ(n log n) in the worst case and Θ(n) when b = O(n/ log n log log n). A minor change determines the optimal reduced isotonic re...

We define a class of algorithms for constructing coresets of (geometric) data sets, and show that algorithms in this class can be dynamized efficiently in the insertiononly (data stream) model. As a result, we show that for a set of points in fixed dimensions, additive and multiplicative ε-coresets for the k-center problem can be maintained in O(1) and O(k) time respectively, using a data struc...

In this paper we study the hardness of the k-Center problem on inputs that model transportation networks. For the problem, an edge-weighted graph G = (V,E) and an integer k are given and a center set C ⊆ V needs to be chosen such that |C| ≤ k. The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and ...

Given a set  of  points in the plane and a constant ,-center problem is to find two closed disks which each covers the whole , the diameter of the bigger one is minimized, and the distance of the two centers is at least . Constrained -center problem is the -center problem in which the centers are forced to lie on a given line . In this paper, we first introduce -center problem and its constrain...

We consider the k-Center problem and some generalizations. For k-Center a set of k center vertices needs to be found in a graph G with edge lengths, such that the distance from any vertex of G to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. [SOD...

We study the LowerBoundedCenter (LBC) problem, which is a clustering problem that can be viewed as a variant of the k-Center problem. In the LBC problem, we are given a set of points P in a metric space and a lower bound λ, and the goal is to select a set C ⊆ P of centers and an assignment that maps each point in P to a center of C such that each center of C is assigned at least λ points. The p...

The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric k-center and an O(log∗(k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must g...