On a Generalization of the p-Center Problem

Abstract We study a generalization of the p-Center Problem, which we call the α-Neighbor p-Center Problem ( p- Center (α) ). Given a complete edge-weighted network, the goal is to minimize the maximum distance of a client to its α nearest neighbors in the set of p centers. We show that in general finding a O(2 poly(¦V¦) )- approximation for p- Center (α) is NP-hard, where ¦V¦ 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 − e) performance guarantee for any fixed e > 0 and any fixed α ⩽ p, unless P = NP. For this case, we present a simple yet efficient algorithm that provides a 4-approximation for α ⩾ 2. If α = 1, our algorithm basically falls back to the algorithm presented in [2] and has a relative performance guarantee of 2.