## 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. 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 (Formula Presented) 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+∈}n approximation scheme for it using Group Steiner Tree techniques.

Original language | English |
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Pages (from-to) | 4-18 |

Number of pages | 15 |

Journal | Theory of Computing Systems |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2016 |

### Bibliographical note

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

Publisher Copyright:

© 2014, Springer Science+Business Media New York.

## Keywords

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