## Abstract

We consider combinatorial algorithms for the minimum-cost flow problem on networks with unit capacities, and special cases of the problem. Historically, researchers have developed special-purpose algorithms that exploit unit capacities. In contrast, for the maximum flow problem, the classical blocking flow and push-relabel algorithms for the general case also have the best bounds known for the special case of unit capacities. In this paper we show that the classical blocking flow push-relabel cost-scaling algorithms of Goldberg and Tarjan (Math. Oper. Res. 15, 430–466, 1990) for general minimum-cost flow problems achieve the best known bounds for unit-capacity problems as well. We also develop a cycle-canceling algorithm that extends Goldberg’s shortest path algorithm (Goldberg SIAM J. Comput. 24, 494–504, 1995) to minimum-cost, unit-capacity flow problems. Finally, we combine our ideas to obtain an algorithm that solves the minimum-cost bipartite matching problem in O(r^{1 / 2}mlog C) time, where m is the number of edges, C is the largest arc cost (assumed to be greater than 1), and r is the number of vertices on the small side of the vertex bipartition. This result generalizes (and simplifies) a result of Duan et al. (2011) and solves an open problem of Ramshaw and Tarjan (2012).

Original language | English (US) |
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Pages (from-to) | 987-1010 |

Number of pages | 24 |

Journal | Theory of Computing Systems |

Volume | 61 |

Issue number | 4 |

DOIs | |

State | Published - Nov 1 2017 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computational Theory and Mathematics

## Keywords

- Algorithms
- Assignment problem
- Bipartite matching
- Minimum-cost flows