A (1 + ln 2) -approximation algorithm for minimum-cost 2-edge-connectivity augmentation of trees with constant radius

We consider the Tree Augmentation problem: given a graph G=(V,E) with edge-costs and a tree T on V disjoint to E, find a minimum-cost edge-subset F⊆E such that T∪F is 2-edge-connected. Tree Augmentation is equivalent to the problem of finding a minimum-cost edge-cover F⊆E of a laminar set-family. The best known approximation ratio for Tree Augmentation is 2, even for trees of radius 2. As laminar families play an important role in network design problems, obtaining a better ratio is a major open problem in connectivity network design. We give a (1+ln2)-approximation algorithm for trees of constant radius. Our algorithm is based on a new decomposition of problem feasible solutions, and on an extension of Steiner Tree technique of Zelikovsky to the Set-Cover problem, which may be of independent interest.