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On the Computational Complexity of Erdős-Szekeres and Related Problems in ℝ3

  • Panos Giannopoulos
  • Christian Knauer
  • Daniel Werner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)

Abstract

The Erdős-Szekeres theorem states that, for every k, there is a number n k such that every set of n k points in general position in the plane contains a subset of k points in convex position. If we ask the same question for subsets whose convex hull does not contain any other point from the set, this is not true: as shown by Horton, there are sets of arbitrary size that do not contain an empty convex 7-gon.

These problems have been studied also from a computational point of view, and, while several polynomial-time algorithms are known for finding a largest (empty) convex subset in the planar case, the complexity of the problems in higher dimensions has been, so far, open. In this paper, we give the first non-trivial results in this direction. First, we show that already in dimension 3 (the decision versions of) both problems are NP-hard. Then, we show that when an empty convex subset is sought, the problem is even W[1]-hard w.r.t. the size of the solution subset.

Keywords

Convex Hull Unit Disk Convex Subset Valid Subset Empty Convex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Panos Giannopoulos
    • 1
  • Christian Knauer
    • 1
  • Daniel Werner
    • 2
  1. 1.Institut für InformatikUniversität BayreuthBayreuthGermany
  2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany

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