Approximation Algorithms for Edge-Dilation k-Center Problems

Abstract

In an ideal point-to-point network, any node would simply choose a path of minimum latency to send packets to any other node; however, the distributed nature and the increasing size of modern communication networks may render such a solution infeasible, as it requires each node to store global information concerning the network. Thus it may be desirable to endow only a small subset of the nodes with global routing capabilites, which gives rise to the following graph-theoretic problem.

Given an undirected graph G = (V,E), a metric l on the edges, and an integer k, a k-center is a set π ⊆ V of size k and an assignment π_v that maps each node to a unique element in Π. We let d_π (u, v) denote the length of the shortest path from utov passing through π_u and π_v and let d _l(u,v) be the length of the shortest u,v-path in G. We then refer to d _π(u,v)/d _l(u,v) as the stretch of the pair (u,v). We let the stretch of a k-center solution Π be the maximum stretch of any pair of nodes u, v ∈ V. The minimum edge-dilation k-center problem is that of finding a k-center of minimum stretch.

We obtain combinatorial approximation algorithms with constant factor performance guarantees for this problem and variants in which the centers are capacitated or nodes may be assigned to more than one center. We also show that there can be no 5/4 - ε approximation for any ε > 0 unless $\mathcal{P} = \mathcal{N}\mathcal{P}$ .

This material is based upon work supported by the National Science Foundation under Grant No. 0105548.