We study the densest subgraph problem and give algorithms via multiplicative weights updated area convexity that converge in $O\left(\frac{\log m}{\epsilon^{2}}\right)$ and $O\left(\frac{\log m}{\epsilon}\right)$ iterations, respectively, both with nearly-linear time per iteration. Compared with the work by Bahmani et al. (2014), our MWU algorithm uses a very different and much simpler procedure for recovering the dense subgraph from the fractional solution and does not employ a binary search. Compared with the work by Boob et al. (2019), our algorithm via area convexity improves the iteration complexity by a factor $\Delta$ -- the maximum degree in the graph, and matches the fastest theoretical runtime currently known via flows (Chekuri et al., 2022) in total time. Next, we study the dense subgraph decomposition problem and give the first practical iterative algorithm with linear convergence rate $O\left(mn\log\frac{1}{\epsilon}\right)$ via accelerated random coordinate descent. This significantly improves over $O\left(\frac{m\sqrt{mn\Delta}}{\epsilon}\right)$ time of the FISTA-based algorithm by Harb et al. (2022). In the high precision regime $\epsilon\ll\frac{1}{n}$ where we can even recover the exact solution, our algorithm has a total runtime of $O\left(mn\log n\right)$, matching the exact algorithm via parametric flows (Gallo et al., 1989). Empirically, we show that this algorithm is very practical and scales to very large graphs, and its performance is competitive with widely used methods that have significantly weaker theoretical guarantees.

翻译：我们研究了稠密子图问题，并提出了通过乘性权重更新的区域凸性算法，其收敛迭代次数分别为 $O\left(\frac{\log m}{\epsilon^{2}}\right)$ 和 $O\left(\frac{\log m}{\epsilon}\right)$，且每次迭代均具有近线性时间复杂度。与 Bahmani 等人（2014）的工作相比，我们的乘性权重更新算法采用了一种截然不同且更为简单的方法从分数解中恢复稠密子图，并且未使用二分搜索。相较于 Boob 等人（2019）的工作，我们基于区域凸性的算法将迭代复杂度改进了 $\Delta$ 倍（$\Delta$ 为图中最大度），并在总时间上匹配了当前通过流方法已知的最快理论运行时间（Chekuri 等人，2022）。接下来，我们研究了稠密子图分解问题，并提出了首个具有线性收敛率 $O\left(mn\log\frac{1}{\epsilon}\right)$ 的实用迭代算法，该算法通过加速随机坐标下降实现。这显著改进了基于 FISTA 的算法（Harb 等人，2022）所需的 $O\left(\frac{m\sqrt{mn\Delta}}{\epsilon}\right)$ 时间。在高精度场景 $\epsilon\ll\frac{1}{n}$ 下（我们甚至能恢复精确解），我们算法的总运行时间为 $O\left(mn\log n\right)$，与基于参数流方法的精确算法（Gallo 等人，1989）相匹配。实验表明，该算法非常实用，可扩展至超大规模图，其性能与广泛使用的、理论保证明显较弱的方法相比具有竞争力。