### Abstract

The minimum-cost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood. Over the last half century many algorithms have been developed to solve the MCF problem and these algorithms have varying worst-case bounds on their running time. However, these worst-case bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is considered the best algorithm in practice and performs best in experimental studies. On the other hand, the Minimum-Mean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies.
To explain these differences in performance in practice we apply the framework of smoothed analysis. For the number of iterations of the MMCC algorithm we show an upper bound of O(m n^2 log(n) log(phi)). Here n is the number of nodes, m is the number of edges, and phi is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of Omega(m log(phi)) for the number of iterations of the MMCC algorithm, which can be strengthened to Omega(m n) when phi=Theta(n^2). For the number of iterations of the NS algorithm we show a smoothed lower bound of Omega(m min\{n,phi\} phi).

Original language | Undefined |
---|---|

Title of host publication | Proceedings of the 21st Computing and Combinatorics Conference (COCOON 2015) |

Editors | Dachuan Xu, Donglei Du, Dingzhu Du |

Place of Publication | Berlin |

Publisher | Springer |

Pages | 701-712 |

Number of pages | 12 |

ISBN (Print) | 978-3-319-21397-2 |

DOIs | |

Publication status | Published - 4 Aug 2015 |

Event | 21st International Conference on Computing and Combinatorics, COCOON 2015 - Beijing, China Duration: 4 Aug 2015 → 6 Aug 2015 Conference number: 21 |

### Publication series

Name | Lecture Notes in Computer Science |
---|---|

Publisher | Springer Verlag |

Volume | 9198 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 21st International Conference on Computing and Combinatorics, COCOON 2015 |
---|---|

Abbreviated title | COCOON |

Country | China |

City | Beijing |

Period | 4/08/15 → 6/08/15 |

### Keywords

- EWI-25942
- IR-96738
- METIS-312555

## Cite this

Cornelissen, K., & Manthey, B. (2015). Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm. In D. Xu, D. Du, & D. Du (Eds.),

*Proceedings of the 21st Computing and Combinatorics Conference (COCOON 2015)*(pp. 701-712). (Lecture Notes in Computer Science; Vol. 9198). Berlin: Springer. https://doi.org/10.1007/978-3-319-21398-9_55