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 ⊆ 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 ∈ 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|) -approximation algorithm for quasi-bipartite instances, when every edge in G has at least one end in 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 ∪{r} .