Approximate Fully Dynamic Directed Densest Subgraph

TL;DR

This paper introduces a new fully dynamic algorithm for maintaining an approximate directed densest subgraph with improved update times, advancing the efficiency of dynamic graph algorithms.

Contribution

It presents the first algorithm achieving near-logarithmic amortized and worst-case update times for maintaining a $(1-\varepsilon)$-approximate directed densest subgraph.

Findings

Achieves $ ilde{O}(rac{ ext{log}^3(n)}{ extvarepsilon^6})$ amortized update time.

Achieves $ ilde{O}(rac{ ext{log}^4(n)}{ extvarepsilon^7})$ worst-case update time.

Improves upon previous algorithms with higher update time complexities.

Abstract

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.