The minimum Manhattan network problem: Approximations and exact solutions

Abstract

Given a set of points in the plane and a constant t ≥ 1, a Euclidean t-spanner is a network in which, for any pair of points, the ratio of the network distance and the Euclidean distance of the two points is at most t. Such networks have applications in transportation or communication network design and have been studied extensively. In this paper we study 1-spanners under the Manhattan (or L 1-) metric. Such networks are called Manhattan networks. A Manhattan network for a set of points is a set of axis-parallel line segments whose union contains an x-and y-monotone path for each pair of points. It is not known whether it is NP-hard to compute minimum Manhattan networks (MMN), i.e. Manhattan networks of minimum total length. In this paper we present an approximation algorithm for this problem. Given a set P of n points, our algorithm computes in O(n log n) time and linear space a Manhattan network for P whose length is at most 3 times the length of an MMN of P. We also establish a mixed-integer programming formulation for the MMN problem. With its help we extensively investigate the performance of our factor-3 approximation algorithm on random point sets.

Minimum Manhattan Network is NP-Complete

“…Although the correctness proof of their algorithm is incomplete [3], the paper showed that checking the existence of a Manhattan path for O(n) specific pairs of points, instead of O(n 2 ) pairs in T × T , is sufficient for determining whether a given network is a Manhattan network. Following this idea, Benkert et al [1,2] proposed an O(n log n)-time 3-approximation algorithm. They also described a mixed-integer programming (MIP) formulation of the MMN problem.…”

“…Most combinatorial constructions [1,2,6,11] rely on the decomposition of the input, by partitioning the plane into several blocks (ortho-convex regions) that can be solved independently. Kato et al [8] presented an O(n 3 )-time 2-approximation algorithm.…”

A Rounding Algorithm for Approximating Minimum Manhattan Networks

“…We continue with the notion of a generating set introduced in [6] and used in approximation algorithms from [1,8]. A generating set is a subset F of pairs of terminals (or, more compactly, of their indices) with the property that a rectilinear network containing l 1 -paths for all pairs in F is a Manhattan network on T. For example, F ∅ consisting of all pairs i, j with R i,j empty is a generating set [6].…”

“…Conversely, let x e , e ∈ E, be a feasible solution for (1). Considering x e 's as capacities of the edges e of Γ, and applying the covering constraints and the Ford-Fulkerson's theorem to each network Γ i,j , (i, j) ∈ − → F , oriented as described below, we conclude the existence in Γ i,j of an integer (t i , t j )-flow of value 1, i.e., of an l 1 -path between t i and t j .…”

“…Let N be a rectilinear network containing l 1 -paths for all pairs in F. To prove that N is a Manhattan network on T , it suffices to establish that for an arbitrary pair {k, k } ∈ F ∅ \ F, the terminals t k and t k are joined in N by an l 1 4 , respectively (minimizing the distance to R k,k in case of ties), we obtain the leftmost vertical strip R i 2 ,i 2 , the lowest horizontal strip R j 1 ,j 1 , and the highest horizontal strip R j 2 ,j 2 crossing the rectangle R k,k . Notice that the strips R j 2 ,j 2 and R i 2 ,i 2 as well as the strips R j 1 ,j 1 and R i 1 ,i 1 constitute crossing configurations.…”