$2$-node-connectivity network design

We consider network design problems in which we are given a graph and seek a min-size $2$-connected subgraph that satisfies a prescribed property. $\bullet$ In the 1-Connectivity Augmentation problem the goal is to augment a connected graph by a min-size edge subset of a specified edge set such that the augmented graph is $2$-connected. We breach the natural ratio of $2$ for this problem and also for the more general Crossing Family Cover problem. $\bullet$ In the $2$-Connected Dominating Set problem we seek a minimum size $2$-connected subgraph that dominates all nodes. We give the first non-trivial approximation algorithm for this problem, with expected ratio $O(σ\log^3 n)$, where $σ=O(\log n \cdot\log\log n\cdot(\log\log\log n)^{3})$. The unifying technique of both results is a reduction to the Subset Steiner Connected Dominating Set problem. Such a reduction was known for edge-connectivity, and we extend it to $2$-node connectivity problems. We show that the same method can be used to obtain easily polylogarithmic approximation ratios that are not too far from the best known ones for several other problems.