Parameterized algorithms for node connectivity augmentation problems

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 $G0=(V,E0)$ by a minimum costs edge set $J$ such that $G0 \cup J$ has higher connectivity than $G0$. In the $k$-Out-Connectivity Augmentation ($k$-OCA) problem, $G0$ is $(k-1)$-out-connected from $s$ and $G0 \cup J$ should be $k$-out-connected from $s$; in the $k$-Connectivity Augmentation ($k$-CA) problem $G0$ is $(k-1)$-connected and $G0 \cup J$ 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 $G0$ is $(k-1)$-edge-connected and $G0 \cup J$ 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 $1.892$.