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Convex transversals
Department of Applied Mathematics and Statistics, Stony Brook University, USA .
Institute of Computer Science, Freie Universität Berlin, Germany .
Institute of Computer Science, Universität Bayreuth, Germany .
Department of Applied Mathematics and Statistics, Stony Brook University, USA .
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2014 (English)In: Computational Geometry, ISSN 0925-7721, Vol. 47, no 2, 224-239 p.Article in journal (Refereed) PublishedText
Abstract [en]

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.

Place, publisher, year, edition, pages
2014. Vol. 47, no 2, 224-239 p.
Keyword [en]
Transversals;Imprecise input;Approximate convexity;Approximate symmetry
National Category
Discrete Mathematics Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-128016DOI: 10.1016/j.comgeo.2012.10.009OAI: oai:DiVA.org:liu-128016DiVA: diva2:928744
Available from: 2016-05-16 Created: 2016-05-16 Last updated: 2016-05-26

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Polishchuk, Valentin
Discrete MathematicsMathematical Analysis

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