## Universal Shortest Paths

- We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.

Author: | Lara Turner, Horst W. Hamacher |
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URN (permanent link): | urn:nbn:de:hbz:386-kluedo-16624 |

Serie (Series number): | Report in Wirtschaftsmathematik (WIMA Report) (128) |

Document Type: | Preprint |

Language of publication: | English |

Year of Completion: | 2010 |

Year of Publication: | 2010 |

Publishing Institute: | Technische Universität Kaiserslautern |

Date of the Publication (Server): | 2010/08/16 |

Tag: | Combinatorial optimization ; shortest path problem ; universal objective function |

Faculties / Organisational entities: | Fachbereich Mathematik |

DDC-Cassification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |

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