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The Complexity of Separating Points in the Plane

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Abstract

We study the following separation problem: given \(n\) connected curves and two points \(s\) and \(t\) in the plane, compute the minimum number of curves one needs to retain so that any path connecting \(s\) to \(t\) intersects some of the retained curves. We give the first polynomial \((\fancyscript{O}(n^3))\) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles.

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Acknowledgments

We would like to thank Primož Škraba for related discussions and the referees for their careful comments.

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Authors and Affiliations

  1. Department of Mathematics, IMFM and FMF, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia

    Sergio Cabello

  2. School of Science and Technology, Middlesex University, The Burroughs, Hendon, London, NW4 4BT, UK

    Panos Giannopoulos

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  1. Sergio Cabello
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  2. Panos Giannopoulos
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Correspondence to Panos Giannopoulos.

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A preliminary version of this work appeared in the Proceedings of the 29th Annual Symposium on Computational Geometry (SoCG), pp. 379–386, 2013. Research by S. Cabello was partially supported by the Slovenian Research Agency, Program P1-0297, Projects J1-4106 and L7-5459, and within the EUROCORES Programme EUROGIGA (Project GReGAS) of the European Science Foundation. Research by P. Giannopoulos was partially supported by the German Science Foundation (DFG) under Grant Kn 591/3-1 while the author was affiliated with Universität Bayreuth, Bayreuth, Germany.

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Cabello, S., Giannopoulos, P. The Complexity of Separating Points in the Plane. Algorithmica 74, 643–663 (2016). https://doi.org/10.1007/s00453-014-9965-6

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  • DOI: https://doi.org/10.1007/s00453-014-9965-6

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