Approximation algorithms for clustering to minimize the sum of diameters

  • Authors:

  • Srinivas Doddi;Madhav V. Marathe;S. S. Ravi;David Scot Taylor;Peter Widmayer

  • Affiliations:

  • Timbre Technologies Freemont, CA;Los Alamos National Laboratory P.O. Box 1663, MS M997 Los Alamos, NM;Department of Computer Science University at Albany - SUNY Albany, NY;Department of Computer Science University of California Los Angeles, CA;Institute for Theoretical Computer Science ETH, 8092 Zürich, Switzerland

  • Venue:
  • Nordic Journal of Computing
  • Year:
  • 2000

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Abstract

We consider the problem of partitioning the n nodes of a complete edge weighted graph into k clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the development of good approximation algorithms. When edge weights satisfy the triangle inequality, we present the first approximation algorithm for the problem. The approximation algorithm yields a solution which has no more than O(k) clusters such that the sum of cluster diameters is within a factor O(ln (n/k)) of the optimal value using exactly k clusters. Our approach also permits a tradeoff among the constant terms hidden by the two big-O terms and the running time. For any fixed k, we present an approximation algorithm that produces k clusters whose total diameter is at most twice the optimal value. When the distances are not required to satisfy the triangle inequality, we show that, unless P = NP, for any ρ ≥ 1, there is no polynomial time approximation algorithm that can provide a performance guarantee of ρ even when the number of clusters is fixed at 3. We also present some results for the problem of minimizing the sum of cluster radii.