On the tree augmentation problem

In the Tree Augmentation problem we are given a tree T = (V, F) and a set E ⊆ V × V of edges with positive integer costs {c_e : E ϵ E}. The goal is to augment T by a minimum cost edge set J ⊆ E such that T [ J is 2-edge-connected. We obtain the following results. Recently, Adjiashvili [SODA 17] introduced a novel LP for the problem and used it to break the 2-Approximation barrier for instances when the maximum cost M of an edge in E is bounded by a constant; his algorithm computes a 1.96418 + ϵ approximate solution in time n(M/-2)O(1) . Using a simpler LP, we achieve ratio 12/7 + ϵ in time 2O(M/-2). This also gives ratio better than 2 for logarithmic costs, and not only for constant costs. In addition, we will show that (for arbitrary costs) the problem admits ratio 3/2 for trees of diameter ≤ 7. One of the oldest open questions for the problem is whether for unit costs (when M = 1) the standard LP-relaxation, so called Cut-LP, has integrality gap less than 2. We resolve this open question by proving that for unit costs the integrality gap of the Cut-LP is at most 28/15 = 2 - 2/15. In addition, we will suggest another natural LP-relaxation that is much simpler than the ones in previous work, and prove that it has integrality gap at most 7/4.