  A graph $G$ is $k$-out-connected from its node $s$ if it contains $k$
internally disjoint $sv$-paths to every node $v$; $G$ is $k$-connected if it is
$k$-out-connected from every node. In connectivity augmentation problems the
goal is to augment a graph $G_0=(V,E_0)$ by a minimum costs edge set $J$ such
that $G_0 \cup J$ has higher connectivity than $G_0$. In the
$k$-Out-Connectivity Augmentation ($k$-OCA) problem, $G_0$ is
$(k-1)$-out-connected from $s$ and $G_0 \cup J$ should be $k$-out-connected
from $s$; in the $k$-Connectivity Augmentation ($k$-CA) problem $G_0$ is
$(k-1)$-connected and $G_0 \cup J$ should be $k$-connected. The parameterized
complexity status of these problems was open even for $k=3$ and unit costs. We
will show that $k$-OCA and $3$-CA can be solved in time $9^p \cdot n^{O(1)}$,
where $p$ is the size of an optimal solution. Our paper is the first that shows
fixed parameter tractability of a $k$-node-connectivity augmentation problem
with high values of $k$. We will also consider the $(2,k)$-Connectivity
Augmentation problem where $G_0$ is $(k-1)$-edge-connected and $G_0 \cup J$
should be both $k$-edge-connected and $2$-connected. We will show that this
problem can be solved in time $9^p \cdot n^{O(1)}$, and for unit costs
approximated within $1.892$.
