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 eE} and edge-costs {c e eE}, source-sink pair s, tV, 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 = (AB, E) and an integer k > 0. The goal is to find a node subset SAB of minimum size |S| such G has k pairwise edge-disjoint paths between SA and SB. 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 : vV}. 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 log3+𝜖 n approximation scheme for it using Group Steiner Tree techniques.

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Notes

  1. 1.

    Being unaware of [3], we derived it independently in an earlier draft [14] of this paper.

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Acknowledgments

We thank Deeparnab Chakrabarty for suggesting us to study the Bipartite Fixed-Cost k-Flow problem. We also thank anonymous referees for useful comments.

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Correspondence to Zeev Nutov.

Additional information

Part of this work was done at DIMACS. We thank DIMACS for their hospitality. A preliminary version appeared in archive [14] in 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.

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Hajiaghayi, M., Khandekar, R., Kortsarz, G. et al. On Fixed Cost k-Flow Problems. Theory Comput Syst 58, 4–18 (2016). https://doi.org/10.1007/s00224-014-9572-6

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Keywords

  • Fixed cost flow
  • Group Steiner tree
  • Network design
  • Approximation algorithms