  In the Tree Augmentation problem we are given a tree $T=(V,F)$ and a set $E
\subseteq V \times V$ of edges with positive integer costs $\{c_e:e \in E\}$.
The goal is to augment $T$ by a minimum cost edge set $J \subseteq E$ such that
$T \cup J$ 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
$1.96418+\epsilon$ approximate solution in time $n^{{(M/\epsilon^2)}^{O(1)}}$.
Using a simpler LP, we achieve ratio $\frac{12}{7}+\epsilon$ in time
$2^{O(M/\epsilon^2)} poly(n)$.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
(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
prove that another natural LP-relaxation, that is much simpler than the ones in
previous work, has integrality gap at most $7/4$.
