NP-completeness and FPT Results for Rectilinear Covering Problems

This paper discusses three rectilinear (that is, axis-parallel) covering problems in d dimensions and their variants. The first problem is the Rectilinear Line Cover where the inputs are n points in Rd and a positive integer k,and we areasked to answer if we can cover these n points with at most k lines where these lines are restricted to be axis parallel. We show that this problem has efficient fixed-parameter tractable (FPT) algorithms. The second problem is the Rectilinear k-Links Spanning Path Problem where the inputs are also n points in Rd and a positive integer k but here we are asked to answer if there is a piecewise linear path through these n points having at most k line-segments (links) where these line-segments are axis- parallel. We prove that this second problem is FPT under the assumption that no two line-segments share the same line. The third problem is the Rectilinear Hyper- plane Cover problem and we are asked to cover a set of n points in d dimensions with k axis-parallel hyperplanes of d - 1 dimensions. We also demonstrate this has an FPT-algorithm. Previous to the results above, only conjectures were enunciated over several years on the NP-completeness of the Rectilinear Minimum Link Traveling Salesman Problem,the Minimum Link Spanning Path Problem and the Recti- linear Hyperplane Cover. We provide the proof that the Rectilinear Minimum Link Traveling Salesman Problem and the Rectilinear Minimum Link Span- ning Path Problem are NP-complete by a reduction from the One-In-Three 3-SAT problem. The NP-completeness of the Rectilinear Hyperplane Cover problem is proved by a reduction from 3-SAT. This suggests dealing with the intractability just discovered with fixed-parameter tractability. Moreover, if we extend our problems to a finite set of orientations, our approach proves these problems remain FPT.