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Well-Separation and Hyperplane Transversals in High Dimensions

Authors Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer , Patrick Schnider



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Helena Bergold
  • Institut für Informatik, Freie Universität Berlin, Germany
Daniel Bertschinger
  • Department of Computer Science, ETH Zürich, Switzerland
Nicolas Grelier
  • Department of Computer Science, ETH Zürich, Switzerland
Wolfgang Mulzer
  • Institut für Informatik, Freie Universität Berlin, Germany
Patrick Schnider
  • Department of Mathematical Sciences, University of Copenhagen, Denmark

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Helena Bergold, Daniel Bertschinger, Nicolas Grelier, Wolfgang Mulzer, and Patrick Schnider. Well-Separation and Hyperplane Transversals in High Dimensions. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 227, pp. 16:1-16:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SWAT.2022.16

Abstract

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • hyperplane transversal
  • high-dimension
  • hardness

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