Node Connectivity Augmentation of Highly Connected Graphs

Node-connectivity augmentation is a fundamental network design problem. We are given a $k$-node connected graph $G$ together with an additional set of links, and the goal is to add a cheap subset of links to $G$ to make it $(k+1)$-node connected. In this work, we characterize completely the computational complexity status of the problem, by showing hardness for all values of $k$ which were not addressed previously in the literature. We then focus on $k$-node connectivity augmentation for $k=n-4$, which corresponds to the highest value of $k$ for which the problem is NP-hard. We improve over the previously best known approximation bounds for this problem, by developing a $\frac{3}{2}$-approximation algorithm for the weighted setting, and a $\frac{4}{3}$-approximation algorithm for the unweighted setting.