We consider the directed Min-Cost Rooted Subset $k$-Edge-Connection problem: given a digraph $G=(V,E)$ with edge costs, a set $T \subseteq V$ of terminals, a root node $r$, and an integer $k$, find a min-cost subgraph of $G$ that contains $k$ edge disjoint $rt$-paths for all $t \in T$. The case when every edge of positive cost has head in $T$ admits a polynomial time algorithm due to Frank [Discret. Appl. Math. 157(6):1242-1254, 2009], and the case when all positive cost edges are incident to $r$ is equivalent to the $k$-Multicover problem. Chan, Laekhanukit, Wei, and Zhang [APPROX/RANDOM, 63:1-63:20, 2020] gave an LP-based $O(\ln k \ln |T|)$-approximation algorithm for quasi-bipartite instances, when every edge in $G$ has an end (tail or head) in $T \cup \{r\}$. We give a simple combinatorial algorithm with the same ratio for a more general problem of covering an arbitrary $T$-intersecting supermodular set function by a minimum cost edge set, and for the case when only every positive cost edge has an end in $T \cup \{r\}$.

翻译：我们考虑有向最小代价根子集$k$-边连接问题：给定一个带边代价的有向图$G=(V,E)$、终端集$T \subseteq V$、根节点$r$及整数$k$，寻找$G$的一个最小代价子图，使得对所有$t \in T$，该子图包含$k$条边不交的$rt$-路。当所有正代价边的头都在$T$中时，该问题存在多项式时间算法（Frank, Discret. Appl. Math. 157(6):1242-1254, 2009）；当所有正代价边均与$r$关联时，该问题等价于$k$-多重覆盖问题。Chan、Laekhanukit、Wei与Zhang [APPROX/RANDOM, 63:1-63:20, 2020]针对拟二分实例（其中$G$中每条边均有一端（尾或头）属于$T \cup \{r\}$）给出了基于线性规划的$O(\ln k \ln |T|)$-近似算法。我们针对以下两个更一般的问题给出了具有相同比率的简单组合算法：用最小代价边集覆盖任意$T$-相交超模集函数的问题，以及仅要求所有正代价边均有一端属于$T \cup \{r\}$的情形。