Convex Transversals

  • Esther M. Arkin
  • Claudia Dieckmann
  • Christian Knauer
  • Joseph S. B. Mitchell
  • Valentin Polishchuk
  • Lena Schlipf
  • Shang Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

Abstract

We answer the question initially posed by Arik Tamir at the Fourth NYU Computational Geometry Day (March, 1987): “Given a collection of compact sets, can one decide in polynomial time whether there exists a convex body whose boundary intersects every set in the collection?”

We prove that when the sets are segments in the plane, deciding existence of the convex stabber is NP-hard. The problem remains NP-hard if the sets are regular polygons. We also show that in 3D the stabbing problem is hard when the sets are balls. On the positive side, we give a polynomial-time algorithm to find a convex transversal of a maximum number of pairwise-disjoint segments in 2D if the vertices of the transversal are restricted to a given set of points. Our algorithm also finds a convex stabber of the maximum number of a set of convex pseudodisks in the plane.

The stabbing problem is related to “convexity” of point sets measured as the minimum distance by which the points must be shifted in order to arrive in convex position; we give a PTAS to find the minimum shift in 2D, and a 2-approximation in any dimension. We also consider stabbing with vertices of a regular polygon – a problem closely related to approximate symmetry detection.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Esther M. Arkin
    • 1
  • Claudia Dieckmann
    • 2
  • Christian Knauer
    • 3
  • Joseph S. B. Mitchell
    • 1
  • Valentin Polishchuk
    • 4
  • Lena Schlipf
    • 2
  • Shang Yang
    • 5
  1. 1.Department of Applied Mathematics and StatisticsStony Brook UniversityUSA
  2. 2.Institute of Computer ScienceFreie Universität BerlinGermany
  3. 3.Institute of Computer ScienceUniversität BayreuthGermany
  4. 4.Helsinki Institute for Information Technology, CS DeptUniversity of HelsinkiFinland
  5. 5.Department of Computer ScienceStony Brook UniversityUSA

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