LP-relaxations for tree augmentation

In the Tree Augmentation problem the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T ∪ F is 2-edge-connected. The best approximation ratio known for the problem is 1.5. In the more general Weighted Tree Augmentation problem, F should be of minimum weight. Weighted Tree Augmentation admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. The problem is equivalent to the problem of covering a laminar set family. Laminar set families play an important role in the design of approximation algorithms for connectivity network design problems. In fact, Weighted Tree Augmentation is the simplest connectivity network design problem for which a ratio better than 2 is not known. Improving this “natural” ratio is a major open problem, which may have implications on many other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for Tree Augmentation. In this paper we give the first LP based approximation for Tree Augmentation with ratio 1.75 < 2. We have two algorithms with ratio 7/4: the first is a primal fitting algorithm and the second is a a dual fitting algorithm. Our main algorithm is the dualfitting one, and it runs in O(mn) time, which is by far faster than all other algorithms with ratio less than 2 for the problem.