Minimum Manhattan Network Problem in Normed Planes with Polygonal Balls: A Factor 2.5 Approximation Algorithm - Catusse, N.; Chepoi, V.; Nouioua, K.; Vaxès, Y. (Algorithmica, 2012)
Let be a centrally symmetric convex polygon of ℝ 2 and ‖ p − q ‖ be the distance between two points p , q ∈ℝ 2 in the normed plane whose unit ball is . For a set T of n points (terminals) in ℝ 2 , a - network on T is a network N ( T )=( V , E ) with the property that its edges are parallel to the directions of and for every pair of terminals t i and t j , the network N ( T ) contains a shortest -path between them, i.e., a path of length ‖ t i − t j ‖. A minimum -network on T is a -network of minimum possible length. The problem of finding minimum -networks has been introduced by Gudmundsson, Levcopoulos, and Narasimhan (APPROX’99) in the case when the unit ball 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 -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.