On the Computational Complexity of Erdős-Szekeres and Related Problems in ℝ3
The Erdős-Szekeres theorem states that, for every k, there is a number nk such that every set of nk 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-hard w.r.t. the size of the solution subset.
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- 1.Avis, D., Rappaport, D.: Computing the largest empty convex subset of a set of points. In: Proceedings of the first Annual Symposium on Computational Geometry, SCG 1985, pp. 161–167. ACM (1985)Google Scholar
- 3.Buchin, K., Plantinga, S., Rote, G., Sturm, A., Vegter, G.: Convex approximation by spherical patches. In: Proceedings of the 23rd EuroCG, pp. 26–29 (2007)Google Scholar
- 5.Chvátal, V., Klincsek, G.: Finding largest convex subsets. In: Congresus Numeratium, pp. 453–460 (1980)Google Scholar
- 8.Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. Springer (2006)Google Scholar
- 10.Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer (2002)Google Scholar