The Minimal Manhattan Network Problem in Three Dimensions

For the Minimal Manhattan Network Problem in three dimensions (MMN3D), one is given a set of points in space, and an admissible solution is an axis-parallel network that connects every pair of points by a shortest path under L 1-norm (Manhattan metric). The goal is to minimize the overall length of the network. Here, we show that MMN3D is $\cal NP$- and $\cal APX$-hard, with a lower bound on the approximability of 1 + 2·10− 5. This lower bound applies to MMN2-3D already, a sub-problem in between the two and three dimensional case. For MMN2-3D, we also develop a 3-approximation algorithm which is the first algorithm for the Minimal Manhattan Network Problem in three dimensions at all.