and

The Tree Augmentation Problem (TAP) is: given a connected graph G = (V, E) and an edge set E on V find a minimum size subset of edges F ⊆ E such that (V, E ∪F) is 2-edge-connected. In the conference version [Even et al. 2001] was sketched a 1.5-approximation algorithm for the problem. Since a full proof was very complex and long, the journal version was cut into two parts. In the first part [Even et al. 2009] was only proved ratio 1.8. An attempt to simplify the second part produced an error in [Even et al. 2011]. Here we give a correct, different, and self contained proof of the ratio 1.5, that is also substantially simpler and shorter than the previous proofs