## 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(√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.

Original language | English |
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Title of host publication | Approximation and Online Algorithms - 11th International Workshop, WAOA 2013, Revised Selected Papers |

Publisher | Springer Verlag |

Pages | 49-60 |

Number of pages | 12 |

ISBN (Print) | 9783319080000 |

DOIs | |

State | Published - 2014 |

Event | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 - Sophia Antipolis, France Duration: 5 Sep 2013 → 6 Sep 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8447 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 11th International Workshop on Approximation and Online Algorithms, WAOA 2013 |
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Country/Territory | France |

City | Sophia Antipolis |

Period | 5/09/13 → 6/09/13 |