Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm

Abstract

Let ${\mathcal{B}}$ be a centrally symmetric convex polygon of ℝ2 and ‖p−q‖ be the distance between two points p,q∈ℝ2 in the normed plane whose unit ball is ${\mathcal{B}}$. For a set T of n points (terminals) in ℝ2, a ${\mathcal{B}}$ -network on T is a network N(T)=(V,E) with the property that its edges are parallel to the directions of ${\mathcal{B}}$ and for every pair of terminals t i and t j , the network N(T) contains a shortest ${\mathcal{B}}$ -path between them, i.e., a path of length ‖t i −t j ‖. A minimum ${\mathcal{B}}$ -network on T is a ${\mathcal{B}}$ -network of minimum possible length. The problem of finding minimum ${\mathcal{B}}$ -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball ${\mathcal{B}}$ is a square (and hence the distance ‖p−q‖ is the l 1 or the l ∞-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (Symposium on Computational Geometry, pp. 393–402, 2009) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4, 3, and 2) for the minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for the minimum ${\mathcal{B}}$ -network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (Chepoi et al. in Theor. Comput. Sci. 390:56–69, 2008, and APPROX-RANDOM, pp. 40–51, 2005) and subsequently used in other factor 2 approximation algorithms for the minimum Manhattan problem.