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Stabbing line segments with disks: Complexity and approximation algorithms
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∈ [ dmin, ηdmax] and some constant η where dmax and dmin 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≥ ηdmax 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. © Springer International Publishing AG 2018- Conference Paper
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- APPROXIMATION ALGORITHMS
- COMPUTATIONAL COMPLEXITY
- CONTINUOUS DISK COVER
- DELAUNAY TRIANGULATIONS
- HITTING SET
- COMPLEX NETWORKS
- COMPUTATIONAL COMPLEXITY
- DRAWING (GRAPHICS)
- GRAPH THEORY
- GRAPHIC METHODS
- IMAGE ANALYSIS
- NETWORK SECURITY
- SURVEYING
- TRIANGULATION
- CONTINUOUS DISK COVER
- DELAU-NAY TRIANGULATIONS
- GEOMETRIC PROBLEMS
- HITTING SETS
- SECURITY APPLICATION
- STRAIGHT-LINE DRAWINGS
- STRAIGHT-LINE SEGMENTS
- UNIFORMLY BOUNDED
- APPROXIMATION ALGORITHMS