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 c...

In this paper we deal with the vertex k-center problem, a problem which is a part of the discrete location theory. Informally, given a set of cities, with intercity distances specified, one has to pick k cities and build warehouses in them so as to minimize the maximum distance of any city from its closest warehouse. We examine several approximation algorithms that achieve approximation factor ...

In the k-center problem, the input is a bound k and n points with the distance between every two of them, such that the distances obey the triangle inequality. The goal is to choose a set of k points to serve as centers, so that the maximum distance from the centers C to any point is as small as possible. This fundamental facility location problem is NP-hard. The symmetric case is well-understo...

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Given a set V of n points and the distances between each pair, the k-center problem asks us to choose a subset C V of size k that minimizes the maximum over all points of the distance from C to the point. This problem is NPhard even when the distances are symmetric and satisfy the triangle inequality, and Hochbaum and Shmoys gave a best-possible 2-approximation for this case. We consider the ve...

In real applications, there are situations where we need to model some problems based on uncertain data. This leads us to define an uncertain model for some classical geometric optimization problems and propose algorithms to solve them. In this paper, we study the k-center problem, for uncertain input. In our setting, each uncertain point Pi is located independently from other points in one of ...

We prove the computational hardness of three k-clustering problems using an (almost) arbitrary Bregman divergence as dissimilarity measure: (a) The Bregman k-center problem, where the objective is to find a set of centers that minimizes the maximum dissimilarity of any input point towards its closest center, and (b) the Bregman k-diameter problem, where the objective is to minimize the maximum ...

In this paper we give approximation algorithms and inapproximability results for various asymmetric k-center with minimum coverage problems. In the k-center with minimum coverage problem, each center is required to serve a minimum number of clients. These problems have been studied by Lim et al. [Theor. Comput. Sci. 2005] in the symmetric setting. In the q-all-coverage k-center problem each cen...

  We consider the maximal covering location-allocation problem with multiple servers. The objective is to maximize the population covered, subject to constraints on the number of service centers, total number of servers in all centers, and the average waiting time at each center. Each center operates as an M/M/k queuing system with variable number of servers. The total costs of establishing cen...

For a set P of n points in R, the Euclidean 2-center problem computes a pair of congruent disks of the minimal radius that cover P . We extend this to the (2, k)-center problem where we compute the minimal radius pair of congruent disks to cover n − k points of P . We present a randomized algorithm with O(nk log n) expected running time for the (2, k)-center problem. We also study the (p, k)-ce...