Approximation algorithms for connectivity augmentation problems

In Connectivity Augmentation problems we are given a graph $H=(V,E_H)$ and an edge set $E$ on $V$, and seek a min-size edge set $J \subseteq E$ such that $H \cup J$ has larger edge/node connectivity than $H$. In the Edge-Connectivity Augmentation problem we need to increase the edge-connectivity by $1$. In the Block-Tree Augmentation problem $H$ is connected and $H \cup S$ should be $2$-connected. In Leaf-to-Leaf Connectivity Augmentation problems every edge in $E$ connects minimal deficient sets. For this version we give a simple combinatorial approximation algorithm with ratio $5/3$, improving the previous $1.91$ approximation that applies for the general case. We also show by a simple proof that if the Steiner Tree problem admits approximation ratio $\alpha$ then the general version admits approximation ratio $1+\ln(4-x)+\epsilon$, where $x$ is the solution to the equation $1+\ln(4-x)=\alpha+(\alpha-1)x$. For the currently best value of $\alpha=\ln 4+\epsilon$ this gives ratio $1.942$. This is slightly worse than the best ratio $1.91$, but has the advantage of using Steiner Tree approximation as a "black box", giving ratio $< 1.9$ if ratio $\alpha \leq 1.35$ can be achieved.