Approximating Minimum Manhattan Networks in Higher Dimensions

Abstract

We consider the minimum Manhattan network problem, which is defined as follows. Given a set of points called terminals in ℝd, 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 \({\cal P}\!=\!{\cal NP}\)). Approximation algorithms are known for 2D, but not for 3D.We present, for any fixed dimension d and any \(\ensuremath{\varepsilon} >0\), an \(O(n^\ensuremath{\varepsilon} )\)-approximation. For 3D, we also give a 4(k − 1)-approximation for the case that the terminals are contained in the union of k ≥ 2 parallel planes.KeywordsApproximation AlgorithmApproximation RatioFull VersionVertical Line SegmentGrid AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.