Improved FPT algorithms for rectilinear k-links spanning path

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.