The Complexity of Geometric Problems in High Dimension

  • Christian Knauer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6108)


Many important NP-hard geometric problems in ℝ d are trivially solvable in time n O(d) (where n is the size of the input), but such a time dependency quickly becomes intractable for higher-dimensional data, and thus it is interesting to ask whether the dependency on d can be mildened. We try to adress this question by applying techniques from parameterized complexity theory.

More precisely, we describe two different approaches to show parameterized intractability of such problems: An “established” framework that gives fpt-reductions from the k-clique problem to a large class of geometric problems in ℝ d , and a different new approach that gives fpt-reductions from the k-Sum problem.

While the second approach seems conceptually simpler, the first approach often yields stronger results, in that it further implies that the d-dimensional problems reduced to cannot be solved in time n o(d), unless the Exponential-Time Hypothesis (ETH) is false.


parameterized complexity geometric dimension lower bounds exponential-time hypothesis 


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Christian Knauer
    • 1
  1. 1.Institut für InformatikUniversität BayreuthGermany

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