On Fixed Cost k -Flow Problems

In the Fixed Cost k -Flow problem, we are given a graph G = ( V , E ) with edge-capacities u e ∣ e ∈ E and edge-costs c e ∣ e ∈ E , source-sink pair s , t ∈ V , and an integer k . The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st -cut in H is at least k . By an approximation-preserving reduction from Group Steiner Tree problem to Fixed Cost k -Flow, we obtain the first polylogarithmic lower bound for the problem; this also implies the first non-constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k -Flow. In the Bipartite Fixed-Cost k -Flow problem, we are given a bipartite graph G = ( A ∪ B , E ) and an integer k > 0. The goal is to find a node subset S ⊆ A ∪ B of minimum size | S | such G has k pairwise edge-disjoint paths between S ∩ A and S ∩ B . We give an O(√(klog k)) approximation for this problem. We also show that we can compute a solution of optimum size with Ω( k /polylog( n )) paths, where n = | A | + | B |. In the Generalized-P2P problem we are given an undirected graph G = ( V , E ) with edge-costs and integer charges b v : v ∈ V . The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [ 11 ]. Besides that, it generalizes many problems such as Steiner Forest, k -Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log 3+ 𝜖 n approximation scheme for it using Group Steiner Tree techniques.