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Stabbing Line Segments with Disks: Complexity and Approximation Algorithms

  • Konstantin KobylkinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10716)

Abstract

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii \(r>0\) where the set of segments forms a straight line drawing \(G=(V,E)\) of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for \(r\in [d_{\min },\eta d_{\max }]\) and some constant \(\eta \) where \(d_{\max }\) and \(d_{\min }\) are Euclidean lengths of the longest and shortest graph edges respectively. Fast \(O(|E|\log |E|)\)-time O(1)-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality \(r\ge \eta d_{\max }\) holds uniformly for some constant \(\eta >0,\) i.e. when lengths of edges of G are uniformly bounded from above by some linear function of r.

Keywords

Computational complexity Approximation algorithms Hitting set Continuous Disk Cover Delaunay triangulations 

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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Mechanics, Ural Branch of RASEkaterinburgRussia
  2. 2.Ural Federal UniversityEkaterinburgRussia

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