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

Jianxin Wang, Peiqiang Tan, Jinyi Yao, Qilong Feng, Jianer Chen

Abstract: The (parameterized) Minimum Link-Length Rectilinear Spanning Path problem in the $\mbi d$ -dimensional Euclidean space $\mbi {\BBR^d}$ ( $\mbi d$ -RSP), for a given set $\mbi S$ of $\mbi n$ points in $\mbi {\BBR^d}$ and a positive integer $\mbi k$ , is to find a piecewise-linear path $\mbi P$ with at most $\mbi k$ line-segments that covers (i.e., contains) all points in $\mbi S$ , where all line-segments in $\mbi P$ 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 $\mbi d$ -RSP problem in which each line-segment $\mbi s$ in the spanning path must cover all the points in the given set $\mbi S$ that share the same line with $\mbi s$ . We present a new parameterized algorithm with running time $\mbi {{O^{\ast}}((2d)^{k})}$ for the constrained $\mbi d$ -RSP problem, which significantly improves the previous best result and is the first parameterized algorithm of running time $\mbi {{O^{\ast}}{(2^{O(k)}})}$ for the constrained $\mbi d$ -RSP problem for a fixed $\mbi d$ . We show that these results can be extended to the Minimum Link-Length Rectilinear Traveling Salesman problem.