  In the Tree Augmentation Problem (TAP) the goal is to augment a tree $T$ by a
minimum size edge set $F$ from a given edge set $E$ such that $T \cup F$ is
$2$-edge-connected. The best approximation ratio known for TAP is $1.5$. In the
more general Weighted TAP problem, $F$ should be of minimum weight. Weighted
TAP admits several $2$-approximation algorithms w.r.t. to the standard cut
LP-relaxation, but for all of them the performance ratio of $2$ is tight even
for TAP. The problem is equivalent to the problem of covering a laminar set
family. Laminar set families play an important role in the design of
approximation algorithms for connectivity network design problems. In fact,
Weighted TAP is the simplest connectivity network design problem for which a
ratio better than $2$ is not known. Improving this "natural" ratio is a major
open problem, which may have implications on many other network design
problems. It seems that achieving this goal requires finding an LP-relaxation
with integrality gap better than $2$, which is a long time open problem even
for TAP. In this paper we introduce such an LP-relaxation and give an algorithm
that computes a feasible solution for TAP of size at most $1.75$ times the
optimal LP value. This gives some hope to break the ratio $2$ for the weighted
case. Our algorithm computes some initial edge set by solving a partial system
of constraints that form the integral edge-cover polytope, and then applies
local search on $3$-leaf subtrees to exchange some of the edges and to add
additional edges. Thus we do not need to solve the LP, and the algorithm runs
roughly in time required to find a minimum weight edge-cover in a general
graph.
