We consider the k-outconnected directed Steiner tree problem (k-DST). Given a directed edge-weighted graph $G=(V,E,w)$, where $V=\{r\}\cup S \cup T$, and an integer $k$, the goal is to find a minimum cost subgraph of $G$ in which there are $k$ edge-disjoint $rt$-paths for every terminal $t\in T$. The problem is know to be NP-hard. Furthermore, the question on whether a polynomial time, subpolynomial approximation algorithm exists for $k$-DST was answered negatively by Grandoni et al. (2018), by proving an approximation hardness of $\Omega (|T|/\log |T|)$ under $NP\neq ZPP$. Inspired by modern day applications, we focus on developing efficient algorithms for $k$-DST in graphs where terminals have out-degree $0$, and furthermore constitute the vast majority in the graph. We provide the first approximation algorithm for $k$-DST on such graphs, in which the approximation ratio depends (primarily) on the size of $S$. We present a randomized algorithm that finds a solution of weight at most $\mathcal O(k|S|\log |T|)$ times the optimal weight, and with high probability runs in polynomial time.

翻译：我们考虑k-出连通有向斯坦纳树问题（k-DST）。给定一个有向边加权图$G=(V,E,w)$，其中$V=\{r\}\cup S \cup T$，以及整数$k$，目标是找到$G$的一个最小成本子图，使得对于每个终端$t\in T$，该子图中存在$k$条边不相交的$rt$路径。已知该问题是NP难的。此外，Grandoni等人（2018）通过证明在$NP\neq ZPP$条件下存在$\Omega (|T|/\log |T|)$的近似难度，否定了k-DST是否存在多项式时间、次多项式近似算法的问题。受现代应用启发，我们专注于在终端出度为$0$且占图中绝大多数的图结构中为k-DST开发高效算法。我们针对此类图提出了首个k-DST近似算法，其近似比主要取决于$S$的规模。我们提出一种随机算法，能以高概率在多项式时间内运行，并找到权重至多为最优权重$\mathcal O(k|S|\log |T|)$倍的解。