On the Fixed Cost k-Flow Problem and related problems

In the Fixed Cost k-Flow problem, we are given a graph G = (V,E) with edge-capacities {ue | e ∈ E} and edge-costs {ce | 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 the Group Steiner on Trees problem is a special case of Fixed Cost k-Flow. This implies the first non constant lower bound for Fixed Cost k-Flow and the first non constant lower bounds for problems that are more general than Fixed Cost kFlow. In particular, the Capacitated Multicommodity Flow and the Capacitated Steiner Network and the Capacitated Buy at Bulk problem. A special case of both Fixed Cost k-Flow and the related Node-Weighted k-Flow problem is the Node-Minimum Bibartite k-Flow problem: given a bipartite graph G = (A ∪ B,E) with edge capacities and an integer k > 0, find a node subset S ⊆ A ∪ B of minimum size |S| such that the minimum capacity of an (S ∩ A,S ∩ B)-cut is at least k. The Node-Weighted k-Flow problem admits an easy O(k)-approximation algorithm, and in [14] is posed an open question whether it admits ratio o(k). We give an O( √ k log k) approximation for Node-Minimum Bibartite k-Flow, which could be a step toward resolving this open question. We also show the following bicriteria result: we can compute a solution of optimum value and deliver Ω(k/polylog|V |) flow. Finally, we give an O(n) ratio for the case of capacities 1 by showing that it is equivalent to the minimization version of the Dense k-subgraph problem. It is widely believed that the minimization version of the Dense k-subgraph problem admits only polynomial ratio. If this is true, then Fixed Cost k-Flow also admits only a polynomial ratio. The final special case of Fixed Cost k-Flow that we study ? Part of this work was done at DIMACS. We thank DIMACS for their hospitality. See also an archive version of the paper: MohammadTaghi Hajiaghayi, Rohit Khandekar, Guy Kortsarz, Zeev Nutov: Combinatorial Algorithms for Capacitated Network Design. CoRR abs/1108.1176 (2011) ?? Supported in part by NSF CAREER award 1053605, ONR YIP award N000141110662, DARPA/AFRL award FA8650-11-1-7162, and University of Maryland Research and Scholarship Award (RASA). The author is also with AT&T Labs– Research, Florham Park, NJ. ? ? ? Supported in part by NSF grant number 434923. is called the Generalized-P2P problem. Besides its practical applications to shift design problems [7], it generalizes many problems such as k-Steiner Tree, Steiner Forest, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a problem related to Buy at Bulk with capacities and with rooted requirements called Connected Rent or Buy Multicommodity Flow. We give a log n approximation scheme for it, using Group Steiner on trees techniques.