Approximating Minimum Manhattan Networks in Higher Dimensions

TL;DR

This paper introduces approximation algorithms for the minimum Manhattan network problem in higher dimensions, providing near-polynomial approximations for fixed dimensions and specialized algorithms for 3D with terminals in parallel planes.

Contribution

It presents the first $O(n^ ext{eps})$-approximation algorithms for fixed dimensions and a new $4(k-1)$-approximation for 3D cases with terminals in parallel planes.

Findings

Achieved $O(n^ ext{eps})$-approximation for fixed dimension $d$

Developed a $4(k-1)$-approximation for 3D with terminals in parallel planes

Established hardness results and limitations for approximation in 2D and 3D

Abstract

We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called \emph{terminals} in $\R^d$, 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 ${\cal P}={\cal NP}$). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension $d$ and any $\eps>0$, an $O(n^\eps)$-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 $k \ge 2$ parallel planes.