Approximation algorithms for connectivity augmentation problems

In Connectivity Augmentation problems we are given a graph $H=(V,EH)$ 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. In the Element Connectivity Augmentation problem we are given a graph $G=(V,E)$, $S \subseteq V$, and connectivity requirements ${r(u,v):u,v \in S}$. The goal is to find a min-size set $J$ of new edges on $S$ such that for all $u,v \in S$ the graph $G \cup J$ contains $r(u,v)$ $uv$-paths such that no two of them have an edge or a node in $V \setminus S$ in common. The problem is NP-hard even when $\max{u,v \in S} r(u,v)=2$. We obtain approximation ratio $3/2$, improving the previous ratio $7/4$.