Approximation Algorithms for Piercing Special Families of Hippodromes: An Extended Abstract

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

Abstract

Polynomial time approximation algorithms are proposed with constant approximation factors for a problem of computing the smallest cardinality set of identical disks whose union intersects each segment from a given set E of n straight line segments on the plane. This problem has important applications in operations research, namely in wireless and road network analysis. It is equivalent to finding the least cardinality piercing (or hitting) set for the corresponding family of n Euclidean r-neighborhoods of straight line segments of E on the plane, which are called r-hippodromes in the literature. When the number of distinct orientations is upper bounded by k of segments from E, a simple $$O(n\log n)$$-time 4k-approximate algorithm is known for this problem. Besides, when E contains arbitrary straight line segments, overlapping at most at their endpoints, $$O(n^4\log n)$$-time 100-approximate algorithm is given recently. In the present paper, simple approximation algorithms are proposed with small approximation factors for E, being edge set of some special plane graphs of interest in road network applications; here the number of distinct orientations of straight line segments from E can be arbitrarily large. More precisely, $$O(n^2)$$-time approximation algorithms are constructed for edge sets of either Gabriel or relative neighborhood graphs or of Euclidean minimum spanning trees with factors of 14, 12 and 10 respectively. These algorithms are much faster, more accurate and conceptually much simpler than the aforementioned 100-approximate algorithm for the general case of the problem on edge sets of arbitrary plane graphs.

Keywords

Operations research Computational geometry Approximation algorithms Geometric piercing problem Straight line segment Hippodrome Proximity graph

References

1. 1.
Agarwal, P., Pan, J.: Near-linear algorithms for geometric hitting sets and set covers. In: Proceedings of the 30th Annual Symposium on Computational Geometry, pp. 271–279. ACM, New York (2014)Google Scholar
2. 2.
Alon, N.: A non-linear lower bound for planar epsilon-nets. Discrete Comput. Geom. 47(2), 235–244 (2011)
3. 3.
Antunes, D., Mathieu, C. and Mustafa, N.: Combinatorics of local search: an optimal 4-local Hall’s theorem for planar graphs. In: Proceedings of the 25th Annual European Symposium on Algorithms, vol. 87, pp. 8:1–8:13. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2017)Google Scholar
4. 4.
Biniaz, A., Liu, P., Maheshwari, A., Smid, M.: Approximation algorithms for the unit disk cover problem in 2D and 3D. Comput. Geom. 60, 8–18 (2017)
5. 5.
Dash, D., Bishnu, A., Gupta, A., Nandy, S.: Approximation algorithms for deployment of sensors for line segment coverage in wireless sensor networks. Wirel. Netw. 19(5), 857–870 (2012)
6. 6.
Efrat, A., Katz, M., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom. 15(4), 215–227 (2000)
7. 7.
Jaromczyk, J., Toussaint, G.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)
8. 8.
Kobylkin, K.: Stabbing line segments with disks: complexity and approximation algorithms. In: van der Aalst, W.M.P., Ignatov, D.I., Khachay, M., Kuznetsov, S.O., Lempitsky, V., Lomazova, I.A., Loukachevitch, N., Napoli, A., Panchenko, A., Pardalos, P.M., Savchenko, A.V., Wasserman, S. (eds.) AIST 2017. LNCS, vol. 10716, pp. 356–367. Springer, Cham (2018).
9. 9.
Kobylkin, K.: Constant factor approximation for intersecting line segments with disks. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P.M. (eds.) LION 12 2018. LNCS, vol. 11353, pp. 447–454. Springer, Cham (2019).
10. 10.
Kobylkin, K.: Approximation algorithms with guaranteed performance for the intersection of edge sets of some metric graphs with equal disks. Trudy Instituta Matematiki i Mekhaniki UrO RAN 25(1), 62–77 (2019)
11. 11.
Matula, D., Sokal, R.: Properties of Gabriel graphs relevant to geographic variation research and the clustering of points in the plane. Geogr. Anal. 12(3), 205–222 (1980)
12. 12.
Madireddy, R. and Mudgal, A.: Stabbing line segments with disks and related problems. In: Proceedings of the 28th Canadian Conference on Computational Geometry, pp. 201–207. Simon Fraser University, Vancouver, Canada (2016)Google Scholar
13. 13.
Mustafa, N., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)
14. 14.
Pyrga, E., Ray, S.: New existence proofs for $$\varepsilon$$-nets. In: Proceedings of the 24th Annual Symposium on Computational Geometry, pp. 199–207. ACM, New York (2008)Google Scholar

© Springer Nature Switzerland AG 2019

Authors and Affiliations

• Konstantin Kobylkin
• 1
• 2
• Irina Dryakhlova
• 2
1. 1.Krasovsky Institute of Mathematics and Mechanics, Ural Branch of RASEkaterinburgRussia
2. 2.Ural Federal UniversityEkaterinburgRussia

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