The Parameterized Complexity of Some Geometric Problems in Unbounded Dimension

  • Panos Giannopoulos
  • Christian Knauer
  • Günter Rote
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5917)

Abstract

We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension d:

i) Given n points in ℝd, compute their minimum enclosing cylinder.

ii) Given two n-point sets in ℝd, decide whether they can be separated by two hyperplanes.

iii) Given a system of n linear inequalities with d variables, find a maximum size feasible subsystem.

We show that (the decision versions of) all these problems are W[1]-hard when parameterized by the dimension d. Our reductions also give a nΩ(d)-time lower bound (under the Exponential Time Hypothesis).

Keywords

parameterized complexity geometric dimension lower bounds minimum enclosing cylinder maximum feasible subsystem 2-linear separability 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Panos Giannopoulos
    • 1
  • Christian Knauer
    • 1
  • Günter Rote
    • 1
  1. 1.Institut für InformatikFreie Universität BerlinBerlinGermany

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