Tight lower bounds for the asymmetric k-center problem

In the k-center problem, the input is a bound k and n points with the distance between every two of them, such that the distances obey the triangle inequality. The goal is to choose a set of k points to serve as centers, so that the maximum distance from the centers C to any point is as small as possible. This fundamental facility location problem is NP-hard. The symmetric case is well-understood from the viewpoint of approximation; it admits a 2–approximation, but not better. We address the approximability of the asymmetric k-center problem. Our first result shows that the linear program used by Archer [Arc01] to devise O(log∗ k)–approximation has integrality ratio that is at least (1 − o(1)) log∗ n; this improves on the previous bound 3 of [Arc01]. Using a similar construction, we then prove that the problem cannot be approximated within a ratio of 1 4 log∗ n, unless NP ⊆ DTIME(n log log n). These are the first lower bounds for this problem that are tight, up to constant factors, with the O(log∗ n)–approximation due to [PV98, Arc01].