### Abstract

Call a bipartite graph G = (X, Y, E) balanced when |X| = |Y|. Given a balanced bipartite graph G with edge costs, the assignment problem asks for a perfect matching in G of minimum total cost. The Hungarian Method can solve assignment problems in time O(mn + n^{2} log n), where n := |X| = |Y| and m := |E|. If the edge weights are integers bounded in magnitude by C > 1, then algorithms using weight scaling, such as that of Gabow and Tarjan, can lower the time to O(m √n log(nC)). There are important applications in which G is unbalanced, with |X| ≠ |Y|, and we require a min-cost matching of size r := min(|X|, |Y|) or, more generally, of some specified size s ≤ r. The Hungarian Method extends easily to find such a matching in time O(ms + s2 log r), but weightscaling algorithms do not extend so easily. We introduce new machinery to find such a matching in time O(m√s log(sC)) via weight scaling. Our results provide some insight into the design space of efficient weight-scaling matching algorithms.

Original language | English (US) |
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Article number | 6375337 |

Pages (from-to) | 581-590 |

Number of pages | 10 |

Journal | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

DOIs | |

State | Published - Dec 1 2012 |

Event | 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States Duration: Oct 20 2012 → Oct 23 2012 |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)