Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

TL;DR

This paper introduces polylogarithmic approximation algorithms for the generalized minimum Manhattan network problem in multiple dimensions, significantly advancing the approximation guarantees for this NP-hard problem.

Contribution

It presents the first polylogarithmic approximation algorithms for GMMN in any dimension, extending known results from 2D to higher dimensions.

Findings

Achieves an $O( ext{log}^{d+1} n)$-approximation for GMMN in $d ext{≥} 2$ dimensions.

Provides an $O( ext{log} n)$-approximation for 2D GMMN.

Shows that the 2D RSA approximation algorithm extends to higher dimensions.

Abstract

Given a set of $n$ terminals, which are points in $d$-dimensional Euclidean space, the minimum Manhattan network problem (MMN) asks for a minimum-length rectilinear network that connects each pair of terminals by a Manhattan path, that is, a path consisting of axis-parallel segments whose total length equals the pair's Manhattan distance. Even for $d=2$, the problem is NP-hard, but constant-factor approximations are known. For $d \ge 3$, the problem is APX-hard; it is known to admit, for any $\eps > 0$, an $O(n^\eps)$-approximation. In the generalized minimum Manhattan network problem (GMMN), we are given a set $R$ of $n$ terminal pairs, and the goal is to find a minimum-length rectilinear network such that each pair in $R$ is connected by a Manhattan path. GMMN is a generalization of both MMN and the well-known rectilinear Steiner arborescence problem (RSA). So far, only special cases of GMMN have been considered. We present an $O(\log^{d+1} n)$-approximation algorithm for GMMN (and, hence, MMN) in $d \ge 2$ dimensions and an $O(\log n)$-approximation algorithm for 2D. We show that an existing $O(\log n)$-approximation algorithm for RSA in 2D generalizes easily to $d>2$ dimensions.