The minimal manhattan network problem in three dimensions

Abstract

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 NP- and APχ-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. © Springer-Verlag Berlin Heidelberg 2009.