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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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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