Approximating the Generalized Minimum Manhattan Network Problem

Abstract

We consider the generalized minimum Manhattan network problem (GMMN). The input to this problem is a set R of n pairs of terminals, which are points in $\mathbb {R}^2$. The goal is to find a minimum-length rectilinear network that connects every pair in R by a Manhattan path, that is, a path of axis-parallel line segments whose total length equals the pair’s Manhattan distance. This problem is a natural generalization of the extensively studied minimum Manhattan network problem (MMN) in which R consists of all possible pairs of terminals. Another important special case is the well-known rectilinear Steiner arborescence problem (RSA). As a generalization of these problems, GMMN is NP-hard. No approximation algorithms are known for general GMMN. We obtain an $O(\log n)$-approximation algorithm for GMMN. Our solution is based on a stabbing technique, a novel way of attacking Manhattan network problems. Some parts of our algorithm generalize to higher dimensions, yielding a simple $O(\log ^{d+1} n)$-approximation algorithm for the problem in arbitrary fixed dimension d. As a corollary, we obtain an exponential improvement upon the previously best $O(n^\varepsilon )$-ratio for MMN in d dimensions (ESA 2011). En route, we show that an existing $O(\log n)$-approximation algorithm for 2D-RSA generalizes to higher dimensions.