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

Abstract

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-approximation algorithm for the minimum-cost strongly connected spanning subgraph problem. For $k=2$, we give a 3-approximation algorithm; this matches the best approximation guarantee known for the special case of the minimum-cost $2$-vertex connected spanning subgraph problem. Besides the standard version, we study another version that is intermediate between the standard version and the problem in its full generality. In the relaxed version of the $(S,T)$ connectivity problem, each edge of positive cost has its head in $T$ but there is no restriction on the tail. We study the relaxed version with the connectivity parameter $k$ fixed at one and observe that this version is at least as hard to approximate as the directed Steiner tree problem. We match this by giving an algorithm that achieves an approximation guarantee of $\alpha(n)+1$ for the relaxed $(S,T)$ connectivity problem, where $\alpha(n)$ denotes the best approximation guarantee available for the directed Steiner tree problem. The key to the analysis is a structural result that decomposes any feasible solution into a set of so-called junction trees that are disjoint on the vertices of $T$. Our algorithm and analysis specialize to the case when the input digraph is acyclic on $T$, meaning that there exists no dicycle that contains two distinct vertices of $T$. In this setting, we show that the relaxed $(S,T)$ connectivity problem is at least as hard to approximate as the set covering problem, and we prove that our algorithm achieves a matching approximation guarantee of $O(\log{|S|})$.