Abstract
Given n points in ℝd and a positive integer k, the Rectilinear k-Links Spanning Path problem is to find a piecewise linear path through these n points having at most k line-segments (Links) where these line-segments are axis-parallel. This problem is known to be NP-complete when d ≥ 3, we first prove that it is also NP-complete in 2-dimensions. Under the assumption that one line-segment in the spanning path covers all the points on the same line, we propose a new FPT algorithm with running time O(d k + 12k k 2 + d k n), which greatly improves the previous best result and is the first FPT algorithm that runs in O *(2O(k)). When d = 2, we further improve this result to O(3.24k k 2 + 1.62k n). For the Rectilinear k-Bends TSP problem, the NP-completeness proof in 2-dimensions and FPT algorithms are also given.
This work is supported by the National Natural Science Foundation of China under Grant (61103033, 61173051), the Doctoral Discipline Foundation of Higher Education Institution of China under Grant (20090162110056).
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References
Lee, D.T., Yang, C.D., Wong, C.K.: Rectilinear Paths among Rectilinear Obstacles. Discrete Applied Mathematics 70(3), 185–215 (1996)
Arkin, E.M., Mitchell, J., Piatko, C.D.: Minimum-link watchman tours. Inf. Process. Lett. 86(4), 203–207 (2003)
Bereg, S., Bose, P., Dumitrescu, A., Hurtado, F., Valtr, P.: Traversing a Set of Points with a Minimum Number of Turns. Discrete & Computational Geometry 41(4), 513–532 (2009)
Lee, D.T., Chen, T.H., Yang, C.: Shortest Rectilinear Paths among Weighted Obstacles. In: Proc. Symposium on Computational Geometry, pp. 301–310 (1990)
Berg, M., Kreveld, M.J., Nilsson, B.J., Overmars, M.H.: Shortest path queries in rectilinear worlds. Int. J. Comput. Geometry Appl. 2(3), 287–309 (1992)
Collins, M.J.: Covering a Set of Points with a Minimum Number of Turns. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 467–474. Springer, Heidelberg (2003)
Arkin, E.M., Bender, M.A., Demaine, E.D., Fekete, S.P., Mitchell, J., Sethia, S.: Optimal Covering Tours with Turn Costs. SIAM J. Comput. 35(3), 531–566 (2005)
Hassin, R., Megiddo, N.: Approximation algorithms for hitting objects with straight lines. Discrete Applied Mathematics 30(1), 29–42 (1991)
Estivill-Castro, V., Heednacram, A., Suraweera, F.: NP-completeness and FPT Results for Rectilinear Covering Problems. J. UCS 16(5), 622–652 (2010)
Kuo, S., Fuchs, W.K.: Efficient spare allocation in reconfigurable arrays. In: DAC, pp. 385–390 (1986)
Kumar, V.S.A., Arya, S., Ramesh, H.: Hardness of Set Cover with Intersection 1. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 624–635. Springer, Heidelberg (2000)
Grantson, M., Levcopoulos, C.: Covering a Set of Points with a Minimum Number of Lines. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 6–17. Springer, Heidelberg (2006)
Wang, J., Li, W., Chen, J.: A parameterized algorithm for the hyperplane-cover problem. Theor. Comput. Sci. 411(44-46), 4005–4009 (2010)
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Wang, J., Yao, J., Feng, Q., Chen, J. (2012). Improved FPT Algorithms for Rectilinear k-Links Spanning Path. In: Agrawal, M., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2012. Lecture Notes in Computer Science, vol 7287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29952-0_52
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DOI: https://doi.org/10.1007/978-3-642-29952-0_52
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