Fault tolerant K-center problems

The basic K-center problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. This problem is known to be NP-hard, and several optimal approximation algorithms that achieve a factor of 2 have been developed for it. We focus our attention on a generalization of this problem, where each vertex is required to have a set of alpha (alpha less than or equal to K) centers close to it. In particular, we study two different versions of this problem. In the first version, each vertex is required to have at least alpha centers close to it. In the second version, each vertex that does not have a center placed on it is required to have at least ct centers close to it. For both these versions we are able to provide polynomial time approximation algorithms that achieve constant approximation factors for any alpha. For the first version we give an algorithm that achieves an approximation factor of 3 for any alpha, and achieves an approximation factor of 2 for alpha < 4. For the second version, we provide algorithms with approximation factors of 2 for any alpha. The best possible approximation factor for even the basic K-center problem is 2. In addition, we give a polynomial time approximation algorithm for a generalization of the K-supplier problem where a subset of at most K supplier nodes must be selected as centers so that every demand node has at least alpha centers close to it. We also provide polynomial time approximation algorithms for all the above problems for generalizations when cost and weight functions are defined on the set of vertices.