## Capacity Inverse Minimum Cost Flow Problem

- Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Author: | Cigdem Güler, Horst Hamacher |
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URN: | urn:nbn:de:hbz:386-kluedo-15097 |

Serie (Series number): | Report in Wirtschaftsmathematik (WIMA Report) (111) |

Document Type: | Preprint |

Language of publication: | English |

Year of Completion: | 2007 |

Year of Publication: | 2007 |

Publishing Institute: | Technische Universität Kaiserslautern |

Date of the Publication (Server): | 2007/12/03 |

Tag: | inverse problems; minimum cost flows; network flows |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 510 Mathematik |

Licence (German): | Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011 |