### Abstract

In the online bipartite matching problem with replacements, all the vertices on one side of the bipartition are given, and the vertices on the other side arrive one by one with all their incident edges. The goal is to maintain a maximum matching while minimizing the number of changes (replacements) to the matching. We show that the greedy algorithm that always takes the shortest augmenting path from the newly inserted vertex (denoted the SAP protocol) uses at most amortized <i>O</i>(log<sup>2</sup> <i>n</i>) replacements per insertion, where <i>n</i> is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for <i>any</i> replacement strategy, almost matching the Ω(log <i>n</i>) lower bound. The previous best strategy known achieved amortized [EQUATION] replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial <i>O</i>(<i>n</i>) bound was known except in special cases. Our analysis immediately implies the same upper bound of <i>O</i>(log<sup>2</sup> <i>n</i>) reassignments for the capacitated assignment problem, where each vertex on the static side of the bipartition is initialized with the capacity to serve a number of vertices.
We also analyze the problem of minimizing the maximum server load. We show that if the final graph has maximum server load <i>L</i>, then the SAP protocol makes amortized <i>O</i>(min{<i>L</i> log<sup>2</sup> <i>n</i>, [EQUATION] reassignments. We also show that this is close to tight because Ω(min{<i>L</i>, [EQUATION] reassignments can be necessary.

Original language | English |
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Title of host publication | Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms |

Publisher | Society for Industrial and Applied Mathematics |

Publication date | 2018 |

Pages | 947-959 |

ISBN (Electronic) | 978-1-61197-503-1 |

DOIs | |

Publication status | Published - 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms - Astor Crowne Plaze, New Orleans French Quarter , New Orleans , United States Duration: 7 Jan 2018 → 10 Jan 2018 Conference number: 29 |

### Conference

Conference | 29th Annual ACM-SIAM Symposium on Discrete Algorithms |
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Number | 29 |

Location | Astor Crowne Plaze, New Orleans French Quarter |

Country | United States |

City | New Orleans |

Period | 07/01/2018 → 10/01/2018 |

Series | Proceedings of the Twenty-ninth Annual Acm-siam Symposium |
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## Cite this

Bernstein, A., Holm, J., & Rotenberg, E. (2018). Online bipartite matching with amortized

*O*(log^{2}*n*) replacements. In*Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms*(pp. 947-959). Society for Industrial and Applied Mathematics. Proceedings of the Twenty-ninth Annual Acm-siam Symposium https://doi.org/10.1137/1.9781611975031.61