Two O(Log*k)-approximation algorithms for the asymmetric k−center problem

Abstract

Given a set V of n points andthe 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 NP-hardev en when the distances are symmetric andsatisfy the triangle inequality, andHo chbaum andShmo ys gave a best-possible 2-approximation for this case. We consider the version where the distances are asymmetric. Panigrahy andVish wanathan gave an O(log*n)-approximation for this case, leading many to believe that a constant approximation factor shouldb e possible. Their approach is purely combinatorial. We show how to use a natural linear programming relaxation to define a promising new measure of progress, anduse it to obtain two different O(log*k)-approximation algorithms. There is hope of obtaining further improvement from this LP, since we do not know of an instance where it has an integrality gap worse than 3.