We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximate directed densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortized time or $\tilde{O}(\log^4(n)/\varepsilon^7)$ worst-case time per edge update (where $\tilde{O}$ hides $\log\log$ factors), based on earlier work by Chekuri and Quanrud [arXiv:2210.02611, arXiv:2310.18146]. This result improves on earlier work done by Sawlani and Wang [arXiv:1907.03037], which guarantees $O(\log^5(n)/\varepsilon^7)$ worst case time for edge insertions and deletions.

翻译：我们提出了一种完全动态算法，用于在边更新时维护一个$(1-\varepsilon)$-近似有向最密子图。基于Chekuri和Quanrud的先前工作[arXiv:2210.02611, arXiv:2310.18146]，该算法在分摊时间复杂度$\tilde{O}(\log^3(n)/\varepsilon^6)$或最坏情况时间复杂度$\tilde{O}(\log^4(n)/\varepsilon^7)$内完成每次边更新（其中$\tilde{O}$隐藏了$\log\log$因子）。这一结果改进了Sawlani和Wang[arXiv:1907.03037]的先前工作，后者在边插入和删除时保证$O(\log^5(n)/\varepsilon^7)$的最坏情况时间。