Computational complexity and approximability are studied for the problem of intersecting of a set of straight line segments with the smallest cardinality set of disks of fixed radii r > 0 where the set of segments forms straight line embedding of possibly non-planar geometric graph. This problem arises in physical network security analysis for telecommunication, wireless and road networks represented by specific geometric graphs defined by Euclidean distances between their vertices (proximity graphs). It can be formulated in a form of known Hitting Set problem over a set of Euclidean r-neighbourhoods of segments. Being of interest computational complexity and approximability of Hitting Set over so structured sets of geometric objects did not get much focus in the literature. Strong NP-hardness of the problem is reported over special classes of proximity graphs namely of Delaunay triangulations, some of their connected subgraphs, half-θ6 graphs and non-planar unit disk graphs as well as APX-hardness is given for non-planar geometric graphs at different scales of r with respect to the longest graph edge length. Simple constant factor approximation algorithm is presented for the case where r is at the same scale as the longest edge length.
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20 October 2016
NUMERICAL COMPUTATIONS: THEORY AND ALGORITHMS (NUMTA–2016): Proceedings of the 2nd International Conference “Numerical Computations: Theory and Algorithms”
19–25 June 2016
Pizzo Calabro, Italy
Research Article|
October 20 2016
Complexity and approximability for a problem of intersecting of proximity graphs with minimum number of equal disks
Konstantin Kobylkin
Konstantin Kobylkin
1Institute of Mathematics and Mechanics,
Ural Branch of RAS
, Sophya Kovalevskaya str., 16/Ekaterinburg, Russia, Graduate School of Economics and Management, Ural Federal University, Mira str., 19/Ekaterinburg, Russia
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URL: [email protected]
AIP Conf. Proc. 1776, 090028 (2016)
Citation
Konstantin Kobylkin; Complexity and approximability for a problem of intersecting of proximity graphs with minimum number of equal disks. AIP Conf. Proc. 20 October 2016; 1776 (1): 090028. https://doi.org/10.1063/1.4965392
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