On The Tree Augmentation Problem

In the TREE AUGMENTATIONproblem we are given a treeT=(V,F)\ and a setE subset of VxVof edges with positive integer costs{ce:e is an element of E}. The goal is to augmentTby a minimum cost edge setJ subset of Esuch that T?Jis 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 of an edge in is bounded by a constant; his algorithm computes a1.96418+epsilon approximate solution in timen(M/epsilon 2)O(1)\. Using a simpler LP, we achieve ratio127+epsilon in time2O(M/epsilon 2)poly(n). This gives ratio better than 2 for logarithmic costs, and not only for constant costs.One of the oldest open questions for the problem is whether for unit costs (whenM=1) the standard LP-relaxation, so calledCut-LP, has integrality gap less than 2. We resolve this open question by proving that for unit costs the integrality gap of theCut-LPis at most28/15=2-2/15. In addition, we will prove that another natural LP-relaxation, that is much simpler than the ones in previous work, has integrality gap at most 7/4.