On Fixed Cost k-Flow Problems

Abstract

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. We show that Group Steiner is a special case of Fixed Cost k -Flow, thus obtaining 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 $O(\sqrt{k\log k})$ 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 [10]. 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 techniques.

Part of this work was done at DIMACS. We thank DIMACS for their hospitality. A preliminary version appeared in archive [13] in 2011.