Approximation algorithm for the kinetic robust K-center problem

Clustering is an important problem and has numerous applications. In this paper we consider an important clustering problem, called the k-center problem. We are given a discrete point set S and a constant integer k, and the goal is to compute a set of k center points to minimize the maximum distance from any point of S to its closest center. We consider both the discrete formulation, in which center points are restricted to be selected from S, and the absolute formulation, in which the centers may be chosen from any point in space. We consider two generalizations of this problem, inspired by issues that arise in real applications. First, we consider a robust version of the problem, in which the user provides a parameter 0 < t ≤ 1, and the algorithm is required to cluster only a fraction t of the points, thus allowing some fraction of outlying points to be ignored. Second, we consider the problem in a kinetic context, where points are assumed to be in motion. We present a kinetic data structure (in the KDS framework) that maintains a (3 + ε)-approximation for the robust discrete k-center problem, and a (4+ ε)-approximation for the robust absolute k-center problem. We also improve on a previous 8-approximation for the non-robust kinetic k-center problem for arbitrary k and show that our data structure supports a (4 + ε)-approximation.