## 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 O(log2 n) replacements per insertion, where n is the total number of vertices inserted. This is the first analysis to achieve a polylogarithmic number of replacements for any replacement strategy, almost matching the (log n) lower bound. The previous best strategy known achieved amortized O(p n) replacements [Bosek, Leniowski, Sankowski, Zych, FOCS 2014]. For the SAP protocol in particular, nothing better than then trivial O(n) bound was known except in special cases. Our analysis immediately implies the same upper bound of O(log2 n) 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 L, then the SAP protocol makes amortized O(min-Llog2 n, p n log n}) reassignments. We also show that this is close to tight because (min-L, p n}) reassignments can be necessary.

Original language | English (US) |
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Title of host publication | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |

Editors | Artur Czumaj |

Publisher | Association for Computing Machinery |

Pages | 947-959 |

Number of pages | 13 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States Duration: Jan 7 2018 → Jan 10 2018 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Other

Other | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 1/7/18 → 1/10/18 |

## ASJC Scopus subject areas

- Software
- General Mathematics

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