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Minimum Manhattan network problem in normed planes with polygonal balls: a factor 2.5 approximation algorithm

Let B be a centrally symmetric convex polygon of R^2 and || p - q || be the distance between two points p,q in R^2 in the normed plane whose unit ball is B. For a set T of n points (terminals) in R^2, a B-Manhattan network on T is a network N(T) = (V,E) with the property that its edges are parallel to the directions of B and for every pair of terminals t_i and t_j, the network N(T) contains a shortest B-path between them, i.e., a path of length || t_i - t_j ||. A minimum B-Manhattan network on T is a B-Manhattan network of minimum possible length. The problem of finding minimum B-Manhattan networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX'99) in the case when the unit ball B is a square (and hence the distance || p - q || is the l_1 or the l_infty-distance between p and q) and it has been shown recently by Chin, Guo, and Sun (SoCG'09) to be strongly NP-complete. Several approximation algorithms (with factors 8, 4 ,3 , and 2) for minimum Manhattan problem are known. In this paper, we propose a factor 2.5 approximation algorithm for minimum B-Manhattan network problem. The algorithm employs a simplified version of the strip-staircase decomposition proposed in our paper (APPROX'05) and subsequently used in other factor 2 approximation algorithms for minimum Manhattan problem.