  The Weighted Tree Augmentation Problem (WTAP) is a fundamental well-studied
problem in the field of network design. Given an undirected tree $G=(V,E)$, an
additional set of edges $L \subseteq V\times V$ disjoint from $E$ called
\textit{links}, and a cost vector $c\in \mathbb{R}_{\geq 0}^L$, WTAP asks to
find a minimum-cost set $F\subseteq L$ with the property that $(V,E\cup F)$ is
$2$-edge connected. The special case where $c_\ell = 1$ for all $\ell\in L$ is
called the Tree Augmentation Problem (TAP). Both problems are known to be
NP-hard.
  For the class of bounded cost vectors, we present a first improved
approximation algorithm for WTAP since more than three decades. Concretely, for
any $M\in \mathbb{Z}_{\geq 1}$ and $\epsilon > 0,$ we present an LP based
$(\delta+\epsilon)$-approximation for WTAP restricted to cost vectors $c$ in
$[1,M]^L$ for $\delta \approx 1.96417$. For the special case of TAP we improve
this factor to $\frac{5}{3}+\epsilon$.
  Our results rely on a new LP, that significantly differs from existing LPs
achieving improved bounds for TAP. We round a fractional solution in two
phases. The first phase uses the fractional solution to decompose the tree and
its fractional solution into so-called $\beta$-simple pairs losing only an
$\epsilon$-factor in the objective function. We then show how to use the
additional constraints in our LP combined with the $\beta$-simple structure to
round a fractional solution in each part of the decomposition.
