### Abstract

In this paper, network flow algorithms for bipartite networks are studied. A network G = (V,E) is called bipartite if its vertex set V can be partitioned into two subsets V_{1} and V_{2} such that all edges have one endpoint in V_{1} and the other in V_{2}. Let n = |V|, n_{1} = |V_{1}|, n_{2} = |V_{2}|, m = |E| and assume without loss of generality that n_{1}≤n_{2}. A bipartite network is called unbalanced if n_{1}≪n_{2} and balanced otherwise. (This notion is necessarily imprecise.) It is shown that several maximum flow algorithms can be substantially sped up when applied to unbalanced networks. The basic idea in these improvements is a two-edge push rule that allows one to `charge' most computation to vertices in V_{1}, and hence develop algorithms whose running times depend on n_{1} rather than n. For example, it is shown that the two-edge push version of Goldberg and Tarjan's FIFO preflow-push algorithm runs in O(n_{1}m+n_{1} ^{3}) time and that the analogous version of Ahuja and Orlin's excess scaling algorithm runs in O(n_{1}m+n_{1} ^{2}log U) time, where U is the largest edge capacity. These ideas are also extended to dynamic tree implementations, parametric maximum flows, and minimum-cost flows.

Original language | English (US) |
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Pages (from-to) | 906-933 |

Number of pages | 28 |

Journal | SIAM Journal on Computing |

Volume | 23 |

Issue number | 5 |

DOIs | |

State | Published - Jan 1 1994 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*23*(5), 906-933. https://doi.org/10.1137/S0097539791199334