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arXiv · 1605.00313

Stabbing line segments with disks: complexity and approximation algorithms

Konstantin Kobylkin
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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 $r\in [d_{\min},ηd_{\max}]$ and some constant $η$ 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 $r\geq ηd_{\max}$ holds uniformly for some constant $η>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$

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BibTeXRIS

Konstantin Kobylkin. 2017-07-20. Stabbing line segments with disks: complexity and approximation algorithms. https://doi.org/10.1007/978-3-319-73013-4_33

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