A Rounding Algorithm for Approximating Minimum Manhattan Networks

Abstract

Abstract. For a set T of n points (terminals) in the plane, a Manhattan network on T is a network N (T ) = (V, E) with the property that its edges are horizontal or vertical segments connecting points in V ⊇ T and for every pair of terminals, the network N (T ) contains a shortest l 1 -path between them. A minimum Manhattan network on T is a Manhattan network of minimum possible length. The problem of finding minimum Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) and its complexity status is unknown. Several approximation algorithms (with factors 8,4, and 3) have been proposed; recently Kato, Imai, and Asano (ISAAC'02) have given a factor 2 approximation algorithm, however their correctness proof is incomplete. In this paper, we propose a rounding 2-approximation algorithm based on a LP-formulation of the minimum Manhattan network problem.

Minimum Manhattan Network is NP-Complete

“…Kato et al [8] presented an O(n 3 )-time 2-approximation algorithm. 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. After that, Chepoi et al [3] used the notion Pareto Envelope and a nice strip-staircase decomposition to divide the plane into several regions, which can be studied individually. Based on this idea, they proposed a 2-approximation rounding algorithm by solving the linear programming relaxation of the MIP.…”

“…Seibert et al [11] proposed a 1.5-approximation algorithm. However, their proof may be incorrect [3]. Muñoz et al [9] gave an NP-hardness proof of this problem in three dimensions and showed that there is no polynomial-time approximation algorithm with a ratio better than 1 + 2 · 10 −5 under the assumption P = NP.…”

“…For a number t ≥ 1, a planar graph G is said to be a t-spanner of a point set T if for all p, q ∈ T , there exists a path in G connecting p and q of length at most t times the Euclidean distance between p and q. The MMN problem for T is exactly the problem to compute the shortest 1-spanner of T under the L 1 -metric [3,5].…”

Minimum Manhattan network is NP-complete

“…For t ≥ 1, a planar graph G is said to be a t-spanner of T if for all p, q ∈ T , there exists a path in G connecting p and q of length at most t times the distance between p and q. The MMN Problem for T is exactly the problem to compute the shortest 1-spanner of T under the L1-norm [3,4].…”