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

• Published in: Algorithmica Vol. 63; no. 1-2; pp. 551 - 567
• Main Authors: Catusse, N., Chepoi, V., Nouioua, K., Vaxès, Y.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.06.2012; Springer; Springer Verlag
• Subjects: Algorithm Analysis and Problem Complexity; Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Computer Science; Computer science; control theory; systems; Computer Systems Organization and Communication Networks; Data Structures and Algorithms; Data Structures and Information Theory; Exact sciences and technology; Flows in networks. Combinatorial problems; Information retrieval. Graph; Mathematics of Computing; Operational research and scientific management; Operational research. Management science; Theoretical computing; Theory of Computation; Geometric network design; Normed plane; Manhattan network; Approximation algorithms; Distance; Probabilistic approach; Shortest path; Distributed system; Approximation algorithm; Terminal; Geometrical model; Computational geometry; NP complete problem; Convex polygon
• ISSN: 0178-4617; 1432-0541
• DOI: 10.1007/s00453-011-9560-z

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.