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We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximatedirected densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortizedtime 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 Chekuriand Quanrud [arXiv:2210.02611, arXiv:2310.18146]. This result improves onearlier 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.

Approximate Fully Dynamic Directed Densest Subgraph