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The Complexity of Geometric Problems in High Dimension

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Theory and Applications of Models of Computation (TAMC 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6108))

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Abstract

Many important NP-hard geometric problems in ℝd 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 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 n o(d), unless the Exponential-Time Hypothesis (ETH) is false.

This research was supported by the German Science Foundation (DFG) under grant Kn 591/3-1.

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Authors and Affiliations

  1. Institut für Informatik, Universität Bayreuth, Germany

    Christian Knauer

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  1. Christian Knauer
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Editors and Affiliations

  1. Charles University, Malostranske nam 25, 11800, Praha 1, Czech Republic

    Jan Kratochvíl , Jiří Fiala  & Petr Kolman ,  & 

  2. State Key Lab. of Computer Science, Institute of Software,, Chinese Academy of Sciences, 100190, Beijing, P.R. China

    Angsheng Li

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Knauer, C. (2010). The Complexity of Geometric Problems in High Dimension. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds) Theory and Applications of Models of Computation. TAMC 2010. Lecture Notes in Computer Science, vol 6108. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13562-0_5

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  • DOI: https://doi.org/10.1007/978-3-642-13562-0_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13561-3

  • Online ISBN: 978-3-642-13562-0

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