The complexity of geometric problems in high dimension

  • Authors:

  • Christian Knauer

  • Affiliations:

  • Institut für Informatik, Universität Bayreuth, Germany

  • Venue:
  • TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
  • Year:
  • 2010

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Abstract

Many important NP-hard geometric problems in ℝd are trivially solvable in time nO(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 ℝd, 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 no(d), unless the Exponential-Time Hypothesis (ETH) is false.