## Abstract

Differentially private algorithms protect individuals in data analysis scenarios by ensuring that there is only a weak correlation between the existence of the user in the data and the result of the analysis. Dynamic graph algorithms maintain the solution to a problem (e.g., a matching) on an evolving input, i.e., a graph where nodes or edges are inserted or deleted over time. They output the value of the solution after each update operation, i.e., continuously. We study (event-level and user-level) differentially private algorithms for graph problems under continual observation, i.e., differentially private dynamic graph algorithms. We present event-level private algorithms for partially dynamic counting-based problems such as triangle count that improve the additive error by a polynomial factor (in the length T of the update sequence) on the state of the art, resulting in the first algorithms with additive error polylogarithmic in T. We also give ε-differentially private and partially dynamic algorithms for minimum spanning tree, minimum cut, densest subgraph, and maximum matching. The additive error of our improved MST algorithm is O(W log
^{3}/
^{2} T/ε), where W is the maximum weight of any edge, which, as we show, is tight up to a (√log T/ε)-factor. For the other problems, we present a partially-dynamic algorithm with multiplicative error (1+β) for any constant β > 0 and additive error O(W log(nW) log(T)/(εβ)). Finally, we show that the additive error for a broad class of dynamic graph algorithms with user-level privacy must be linear in the value of the output solution's range.

Original language | English |
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Title of host publication | 29th Annual European Symposium on Algorithms, ESA 2021 |

Editors | Petra Mutzel, Rasmus Pagh, Grzegorz Herman |

ISBN (Electronic) | 9783959772044 |

DOIs | |

Publication status | Published - 1 Sep 2021 |

Event | European Symposium on Algorithms (ESA) - Online, Portugal Duration: 6 Sep 2021 → 8 Sep 2021 http://algo2021.tecnico.ulisboa.pt/ESA2021/ http://algo2021.tecnico.ulisboa.pt/ESA2021/index.html |

### Conference

Conference | European Symposium on Algorithms (ESA) |
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Country/Territory | Portugal |

Period | 6/09/21 → 8/09/21 |

Internet address |

## Austrian Fields of Science 2012

- 102031 Theoretical computer science

## Keywords

- Continual observation
- Differential privacy
- Dynamic graph algorithms