## Abstract

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum outdegree is proportional to the arboricity α of the graph, in, either, an amortised update time of O(log^{2} nlog α), or a worst-case update time of O(log^{3} nlog α). On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either O(log nlog α), amortised, or O(log^{2} nlog α), worst-case, for the problem of maintaining an edge-orientation with at most O(α + log n) out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in n and α. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a (1 + ε) approximation of the maximum subgraph density, ρ, of the dynamic graph. Our algorithms have update times of O(ε^{−6} log^{3} nlog ρ) worst-case, and O(ε^{−4} log^{2} nlog ρ) amortised, respectively. We may output a subgraph H of the input graph where its density is a (1 + ε) approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the O(ε^{−6} log^{4} n) algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an O(ε^{−6} log^{3} nlog α) worst-case update time algorithm for maintaining a (1 + ε)OPT + 2 approximation of the optimal out-orientation of a graph with adaptive arboricity α, improving the O(ε^{−6}α^{2} log^{3} n) algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worst-case polylogarithmic dynamic algorithm for decomposing into O(α) forests. Thirdly, we obtain arboricity-adaptive fully-dynamic deterministic algorithms for a variety of problems including maximal matching, ∆ + 1 colouring, and matrix vector multiplication. All update times are worst-case O(α + log^{2} nlog α), where α is the current arboricity of the graph. For the maximal matching problem, the state-of-the-art deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time O(α^{2} + log^{2} n), and by Neiman and Solomon from STOC 2013 runs in time O(√m). We give improved running times whenever the arboricity α ∈ ω(log n√log log n).

Original language | English (US) |
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Pages | 3062-3088 |

Number of pages | 27 |

DOIs | |

State | Published - 2024 |

Externally published | Yes |

Event | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 - Alexandria, United States Duration: Jan 7 2024 → Jan 10 2024 |

### Conference

Conference | 35th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2024 |
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Country/Territory | United States |

City | Alexandria |

Period | 1/7/24 → 1/10/24 |

## ASJC Scopus subject areas

- Software
- General Mathematics