License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.SWAT.2022.16
URN: urn:nbn:de:0030-drops-161766
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Bergold, Helena ; Bertschinger, Daniel ; Grelier, Nicolas ; Mulzer, Wolfgang ; Schnider, Patrick

Well-Separation and Hyperplane Transversals in High Dimensions

LIPIcs-SWAT-2022-16.pdf (0.7 MB)


A family of k point sets in d dimensions is well-separated if the convex hulls of any two disjoint subfamilies can be separated by a hyperplane. Well-separation is a strong assumption that allows us to conclude that certain kinds of generalized ham-sandwich cuts for the point sets exist. But how hard is it to check if a given family of high-dimensional point sets has this property? Starting from this question, we study several algorithmic aspects of the existence of transversals and separations in high-dimensions.
First, we give an explicit proof that k point sets are well-separated if and only if their convex hulls admit no (k - 2)-transversal, i.e., if there exists no (k - 2)-dimensional flat that intersects the convex hulls of all k sets. It follows that the task of checking well-separation lies in the complexity class coNP. Next, we show that it is NP-hard to decide whether there is a hyperplane-transversal (that is, a (d - 1)-transversal) of a family of d + 1 line segments in ℝ^d, where d is part of the input. As a consequence, it follows that the general problem of testing well-separation is coNP-complete. Furthermore, we show that finding a hyperplane that maximizes the number of intersected sets is NP-hard, but allows for an Ω((log k)/(k log log k))-approximation algorithm that is polynomial in d and k, when each set consists of a single point. When all point sets are finite, we show that checking whether there exists a (k - 2)-transversal is in fact strongly NP-complete.
Finally, we take the viewpoint of parametrized complexity, using the dimension d as a parameter: given k convex sets in ℝ^d, checking whether there is a (k-2)-transversal is FPT with respect to d. On the other hand, for k ≥ d+1 finite point sets in ℝ^d, it turns out that checking whether there is a (d-1)-transversal is W[1]-hard with respect to d.

BibTeX - Entry

  author =	{Bergold, Helena and Bertschinger, Daniel and Grelier, Nicolas and Mulzer, Wolfgang and Schnider, Patrick},
  title =	{{Well-Separation and Hyperplane Transversals in High Dimensions}},
  booktitle =	{18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-236-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{227},
  editor =	{Czumaj, Artur and Xin, Qin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-161766},
  doi =		{10.4230/LIPIcs.SWAT.2022.16},
  annote =	{Keywords: hyperplane transversal, high-dimension, hardness}

Keywords: hyperplane transversal, high-dimension, hardness
Collection: 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)
Issue Date: 2022
Date of publication: 22.06.2022

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