## Complexity and Approximability of the Maximum Flow Problem with Minimum Quantities

- We consider the maximum flow problem with minimum quantities (MFPMQ), which is a variant of the maximum flow problem where the flow on each arc in the network is restricted to be either zero or above a given lower bound (a minimum quantity), which may depend on the arc. This problem has recently been shown to be weakly NP-complete even on series-parallel graphs. In this paper, we provide further complexity and approximability results for MFPMQ and several special cases. We first show that it is strongly NP-hard to approximate MFPMQ on general graphs (and even bipartite graphs) within any positive factor. On series-parallel graphs, however, we present a pseudo-polynomial time dynamic programming algorithm for the problem. We then study the case that the minimum quantity is the same for each arc in the network and show that, under this restriction, the problem is still weakly NP-complete on general graphs, but can be solved in strongly polynomial time on series-parallel graphs. On general graphs, we present a \((2 - 1/\lambda) \)-approximation algorithm for this case, where \(\lambda\) denotes the common minimum quantity of all arcs.

Author: | Clemens Thielen, Stephan Westphal |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-31819 |

Serie (Series number): | Report in Wirtschaftsmathematik (WIMA Report) (143) |

Document Type: | Article |

Language of publication: | English |

Publication Date: | 2012/07/05 |

Year of Publication: | 2012 |

Publishing Institute: | Technische Universität Kaiserslautern |

Date of the Publication (Server): | 2012/07/06 |

Number of page: | 13 |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 519 Wahrscheinlichkeiten, angewandte Mathematik |

MSC-Classification (mathematics): | 90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C27 Combinatorial optimization |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vom 02.07.2012 |