On a Generalization of the p-Center Problem

We study a generalization of the p{Center Problem, which we call the {Neighbor p{Center Problem (p{Center ()). Given a complete edge{ weighted network, the goal is to minimize the maximum distance of a client to its nearest neighbors in the set of p centers. We show that in general nding a O(2 poly(jV j)){approximation for p{ Center () is NP{hard, where jV j denotes the number of nodes in the network. If the distances are required to satisfy the triangle inequality, there can be no polynomial time approximation algorithm with a (2 ? ") performance guarantee for any xed " > 0 and any xed p, unless P = NP. For this case, we present a simple yet eecient algorithm that provides a 4{approximation for 2. If = 1, our algorithm basically falls back to the algorithm presented in 2] and has a relative performance guarantee of 2.