## Departmental Papers (CIS)

#### Date of this Version

June 2004

#### Document Type

Conference Paper

#### Recommended Citation

Julia Chuzhoy, Sudipto Guha, Eran Halperin, Sanjeev Khanna, Guy Kortsarz, Robert Krauthgamer, and Joseph Naor, "Asymmetric *k*-Center is log^{*}* n*-Hard to Approximate", . June 2004.

#### Abstract

In the Asymmetric *k*-Center problem, the input is an integer *k* and a complete digraph over *n* points together with a distance function obeying the directed triangle inequality. The goal is to choose a set of *k* points to serve as centers and to assign all the points to the centers, so that the maximum distance of any point to its center is as small as possible.

We show that the Asymmetric *k*-Center problem is hard to approximate up to a factor of log^{*} *n* - *O*(1) unless **NP** is a subset of or equal to **DTIME**(*n*^{log log n}). Since an *O*(log^{*} *n*)-approximation algorithm is known for this problem, this resolves the asymptotic approximability of this problem. This is the first natural problem whose approximability threshold does not polynomially relate to the known approximation classes. We also resolve the approximability threshold of the metric (symmetric) *k*-Center problem with costs.

#### Keywords

approximation algorithms, asymmetric k-center, hardness of approximation, metric k-center

**Date Posted:** 23 September 2004

This document has been peer reviewed.

## Comments

Proceedings of the 36th Annual ACM Symposium on Theory of Computing, 2004, pages 21-27.

Publisher URL: http://doi.acm.org/10.1145/1007352.1007363