On rooted $k$-connectivity problems in quasi-bipartite digraphs

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 $T \cup \{r\}$.