### Abstract

We review a class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation. All the methods in the class use the notion of ε-complementary slackness, and most do not explicitly manipulate any "global" objects such as paths, trees, or cuts. Interestingly, these methods have stimulated a large number of new serial computational complexity results. We develop the basic theory of these methods and present two specific methods, the ε-relaxation algorithm for the minimum-cost flow problem, and the auction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N^{3} log NC) and O(NA log NC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implement ε-relaxation in a completely asynchronous, "chaotic" environment in which some processors compute faster than others, some processors communicate faster than others, and there can be arbitrarily large communication delays.

Original language | English (US) |
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Pages (from-to) | 203-243 |

Number of pages | 41 |

Journal | Mathematical Programming |

Volume | 42 |

Issue number | 1-3 |

DOIs | |

State | Published - Apr 1 1988 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Software
- Mathematics(all)

### Keywords

- Network flows
- asynchronous algorithms
- complexity
- distributed algorithms
- relaxation

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## Cite this

*Mathematical Programming*,

*42*(1-3), 203-243. https://doi.org/10.1007/BF01589405