Approximating Minimum Manhattan Networks in Higher Dimensions

Abstract

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in ${\mathbb {R}}^{d}$ , find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, L 1-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\mathcal{P}} = \mathcal{NP}$ ). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension d and any ?>0, an O(n ? )-approximation algorithm. For 3D, we also give a 4(k?1)-approximation algorithm for the case that the terminals are contained in the union of k?2 parallel planes.