On Rooted k-Connectivity Problems in Quasi-Bipartite Digraphs

We consider the directed Min-Cost Rooted Subset k -Edge-Connection problem: given a digraph $$G=(V,E)$$ G = ( V , E ) with edge costs, a set $$T \subseteq V$$ T ⊆ 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 $$t \in T$$ t ∈ 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 et al. (APPROX/RANDOM, 2020) gave an LP-based $$O(\ln k \ln |T|)$$ O ( ln k ln | T | ) -approximation algorithm for quasi-bipartite instances, when every edge in G has at least one end in $$T \cup \{r\}$$ T ∪ { r } . We give a simple combinatorial algorithm with the same approximation ratio for a more general problem of covering an arbitrary T-intersecting supermodular set function by a min-cost edge set, and for the case when only every positive cost edge has at least one end in $$T \cup \{r\}$$ T ∪ { r } .