## 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 \(O(\sqrt {k\log k})\) 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 [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.

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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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## 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