Approximation Algorithms for Minimum-Cost $k\hbox{-}(S, T)$ Connected Digraphs

In the minimum-cost $k\hbox{-}(S,T)$ connected digraph (abbreviated as $k\hbox{-}(S,T)$ connectivity) problem we are given a positive integer $k$, a directed graph $G=(V,E)$ with nonnegative costs on the edges, and two subsets $S,T$ of $V$; the goal is to find a subset of edges $\widehat{E}$ of minimum cost such that the subgraph $(V,\widehat{E})$ has $k$ edge-disjoint directed paths from each vertex in $S$ to each vertex in $T$. Most of our results focus on a specialized version of the problem that we call the standard version, where every edge of positive cost has its tail in $S$ and its head in $T$. This version of the problem captures NP-hard problems such as the minimum-cost $k$-vertex connected spanning subgraph problem. We give an approximation algorithm with a guarantee of $O((\log{k})(\log{n}))$ for the standard version of the $k\hbox{-}(S,T)$ connectivity problem, where $n$ denotes the number of vertices. For $k=1$, we give a simple 2-approximation algorithm that generalizes a well-known 2-appro...