On rooted k-connectivity problems in quasi-bipartite digraphs
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We consider the directed Rooted Subset k-Edge-Connectivity problem: given a set T ⊆ V of terminals in a digraph G=(V+r,E) with edge costs 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, and the case when all positive cost edges are incident to r is equivalent to the k-Multicover problem. Recently, [Chan et al. APPROX20] obtained ratio O(ln k ln |T|) for quasi-bipartite instances, when every edge in G has an end in T+r. We give a simple proof for 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+r.
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