Abstract
An abstract NP-hard covering problem is presented and fixed-parameter tractable algorithms for this problem are described. The running times of the algorithms are expressed in terms of three parameters: n, the number of elements to be covered, k, the number of sets allowed in the covering, and d, the combinatorial dimension of the problem. The first algorithm is deterministic and has running time O’(k dk n). The second algorithm is also deterministic and has running time O’(k d(k+1)+n d+1). The third is a Monte-Carlo algorithm that runs in time O’(k d(k+1)+c2d k⌈d+1/2⌉⌊d+1)/2⌋ n log n time and is correct with probability 1-n -c. Here, the O’ notation hides factors that are polynomial in d. These algorithms lead to fixed-parameter tractable algorithms for many geometric and non-geometric covering problems.
This research was partly funded by NSERC, MITACS, FCAR and CRM.
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References
P.K. Agarwal and C. M. Procopiuc. Exact and approximation algorithms for clustering. In Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms (SODA 1998), pages 658–667, 1998.
George E. Andrews. The Theory of Partitions. Addison-Wesley, 1976.
B. Brodén, M. Hammar, and B. J. Nilsson. Guarding lines and 2-link polygons is APX-hard. In Proceedings of the 13th Canadian Conference on Computational Geometry, pages 45–48, 2001.
J. Chen, I. Kanj, and W. Jia. Vertex cover: further observations and further improvements. Journal of Algorithms, 41:280–301, 2001.
R.G. Downey and M.R. Fellows. Parameterized Complexity. Springer, 1999.
Leonhardo Eulero. Introductio in Analysin Infinitorum. Tomus Primus, Lausanne, 1748.
L. J. Guibas, M. H. Overmars, and J.-M. Robert. The exact fitting problem for points. Computational Geometry: Theory and Applications, 6:215–230, 1996.
R. Hassin and N. Megiddo. Approximation algorithms for hitting objects by straight lines. Discrete Applied Mathematics, 30:29–42, 1991.
D. S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer Systems Sciences, 9:256–278, 1974.
R. M. Karp. Reducibility Among Combinatorial Problems, pages 85–103. Plenum Press, 1972.
V. S. A. Kumar, S. Arya, and H. Ramesh. Hardness of set cover with intersection 1. In Proceedings of the 27th International Colloquium on Automata, Languages and Programming, pages 624–635, 2000.
N. Megiddo and A. Tamir. On the complexity of locating linear facilities in the plane. Operations Research Letters, 1:194–197, 1982.
D. Nussbaum. Rectilinear p-piercing problems. In ISSAC’ 97. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, July 21–23, 1997, Maui, Hawaii, pages 316–323, 1997.
Franco P. Preparata and Michael Ian Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985.
M. Sharir and E. Welzl. Rectilinear and polygonal p-piercing and p-center problems. In Proceedings of the Twelfth Annual Symposium On Computational Geometry (ISG’ 96), pages 122–132, 1996.
V. N. Vapnik and A. Y. A. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16:264–280, 1971.
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Langerman, S., Morin, P. (2002). Covering Things with Things. In: Möhring, R., Raman, R. (eds) Algorithms — ESA 2002. ESA 2002. Lecture Notes in Computer Science, vol 2461. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45749-6_58
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DOI: https://doi.org/10.1007/3-540-45749-6_58
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