Covering a laminar family by leaf to leaf links

The Tree Augmentation Problem (TAP) is: given a tree T = (V, E) and a set E of edges (called links) on V disjoint to E , find a minimum size edge subset F ⊆ E so that T +F is 2-edge-connected. TAP is equivalent to the problem of finding a minimum size edge-cover F ⊆ E of a laminar set-family. We consider the restriction LL-TAP of TAP to instances when every link in E connects two leaves of T . The best approximation ratio for TAP is 3/2 [3, 4, 5], and no better ratio was known for LL-TAP. All the previous approximation algorithms that achieve a ratio better than 2 for TAP, or even for LL-TAP, were quite involved. For LL-TAP we obtain the following approximation ratios: 17/12 for general trees, 11/8 for trees of height 3, and 4/3 for trees of height 2. We also give a very simple 3/2-approximation algorithm (for general trees) and prove that it computes a solution of size at most min{ 3 2 t, 5 3 t}, where t is the minimum size of an edge-cover of the leaves, and t is the optimal value of the natural LP-relaxation for the problem of covering the leaf edges only. This provides the first evidence that the integrality gap of a natural LP-relaxation for LL-TAP is less than 2.