## Abstract

We study a type of network ow problem that we call the minimum-cost F-graph flow problem. This problem generalizes the typical minimum-cost network flow problem by allowing the underlying network to be a directed hypergraph rather than a directed graph. This new problem is pertinent because it can be used to model network ow problems that occur in a dynamic, stochastic environment. We formulate this problem as an integer program, and we study specifically the case where every node has at least one outgoing edge with no capacity constraint. We show that even with this restriction, the problem of finding an integral solution is NP-hard. However, we can show that all of the inequality constraints of our formulation are either facet-defining or redundant.

Original language | English (US) |
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Pages (from-to) | 2355-2378 |

Number of pages | 24 |

Journal | SIAM Journal on Optimization |

Volume | 30 |

Issue number | 3 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020 Society for Industrial and Applied Mathematics.

## Keywords

- Directed hypergraphs
- F-graphs
- Facet-defining inequalities
- Network ow
- Stochastic integer programming