Geometric clustering: fixed-parameter tractability and lower bounds with respect to the dimension

  • Authors:

  • Sergio Cabello;Panos Giannopoulos;Christian Knauer;Günter Rote

  • Affiliations:

  • University of Ljubljana, IMFM and FMF, SI, Ljubljana, Slovenia;Humboldt-Universität zu Berlin, Institut für Informatik, Berlin, Germany;Freie Universität Berlin, Institut für Informatik, Berlin, Germany;Freie Universität Berlin, Institut für Informatik, Berlin, Germany

  • Venue:
  • Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
  • Year:
  • 2008

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Abstract

We present an algorithm for the 3-center problem in (ℝd, L∞), i.e., for finding the smallest side length for 3 cubes that cover a given n-point set in ℝd, that runs in O(n log n) time for any fixed dimension d. This shows that the problem is fixed-parameter tractable when parameterized with d. On the other hand, using tools from parameterized complexity theory, we show that this is unlikely to be the case with the k-center problem in (ℝd, L2), for any k ≥ 2. In particular, we prove that deciding whether a given n-point set in ℝd can be covered by the union of 2 balls of given radius is W[1]-hard with respect to d, and thus not fixed-parameter tractable unless FPT=W[1]. Our reduction also shows that even an O(no(d))-time alorithm for the latter does not exist, unless SNP ⊆ DTIME(2o(n)).