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)


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.


Convex Hull Unit Disk Convex Subset Valid Subset Empty Convex 
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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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