A rounding algorithm for approximating minimum Manhattan networks

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 l1-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