Connectivity augmentation is fixed-parameter tractable

In the vertex connectivity augmentation problem, we are given an undirected $n$-vertex graph $G$, a set of links $L \subseteq \binom{V(G)}{2} \setminus E(G)$, and integers $\lambda$ and $k$. The task is to insert at most $k$ links from $L$ to $G$ to make $G$ $\lambda$-vertex-connected. We show that the problem is fixed-parameter tractable (FPT) when parameterized by $\lambda$ and $k$, by giving an algorithm with running time $2^{O(k \log (k + \lambda))} n^{O(1)}$. This improves upon a recent result of Carmesin and Ramanujan [SODA'26], who showed that the problem is FPT parameterized by $k$ but only when $\lambda \le 4$. We also consider the analogous edge connectivity augmentation problem, where the goal is to make $G$ $\lambda$-edge-connected. We show that the problem is FPT when parameterized by $k$ only, by giving an algorithm with running time $2^{O(k \log k)} n^{O(1)}$. Previously, such results were known only under additional assumptions on the edge connectivity of $G$.