On the Minimum Link-Length Rectilinear Spanning Path Problem: Complexity and Algorithms

The (parameterized) Minimum Link-Length Rectilinear Spanning Path problem in the -ddimensional Euclidean space R ( -RSP), for a given set of points in R and a positive integer , is to find a piecewise-linear path with at most k line-segments that covers (i.e., contains) all points in , where all line-segments in are axis-parallel. We first prove that the problem 2-RSP is NP-complete, improving the previously known result that the problem 10-RSP is NP-complete. We then consider a constrained d-RSP problem in which each line-segment in the spanning path must cover all the points in the given set that share the same line with . We present a new parameterized algorithm with running time O*((2d )^k) for the constrained -RSP problem, which significantly improves the previous best result and is the first parameterized algorithm of running time O*(2^O(k)) for the constrained d-RSP problem for a fixed . We show that these results can be extended to the Minimum Link-Length Rectilinear Traveling Salesman problem.