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

  • Authors:

  • Julia Chuzhoy;Sudipto Guha;Eran Halperi;Sanjeev Khanna;Guy Kortsarz;Joseph (Seffi) Nao

  • Affiliations:

  • Technion, Haifa, Israel;University of Pennsylvania, Philadelphia, PA;UC Berkeley, Berkeley, CA;University of Pennsylvania, Philadelphia, PA;Rutgers University, Camden, NJ;Technion, Haifa, Israel

  • Venue:
  • STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
  • Year:
  • 2004

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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 - Θ(1) unless NP ⊆ DTIME(nlog log n). Since an O(log* n)-approximation algorithm is known for this problem, this essentially resolves the 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 k-Center problem with costs.