Asymmetric k-center is log* n-hard to approximate

  • Authors:

  • Julia Chuzhoy;Sudipto Guha;Eran Halperin;Sanjeev Khanna;Guy Kortsarz;Robert Krauthgamer;Joseph (Seffi) Naor

  • Affiliations:

  • MIT, Cambridge Massachusetts;University of Pennsylvania, Philadelphia, Pennsylvania, Philadelphia, PA;International Computer Science Institute, Berkeley, California, and University of California, Berkeley, Berkeley, California, CA;University of Pennsylvania, Philadelphia, Pennsylvania, Philadelphia, PA;Rutgers University, Camden, New Jersey;International Computer Science Institute, Berkeley, California, and University of California, Berkeley, Berkeley, California, CA;Technion, Haifa, Israel

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 2005

Quantified Score

Hi-index 0.04

Visualization

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 from 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 ⊆ DTIME(nlog log n). Since an O(log*n)-approximation algorithm is known for this problem, this resolves the asymptotic approximability of ASYMMETRIC k-CENTER. 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.