Multiplicative Weights Update, Area Convexity and Random Coordinate Descent for Densest Subgraph Problems

We study the densest subgraph problem and give algorithms via multiplicative weights update and area convexity that converge in Oleft(fraclog mε2right) and Oleft(fraclog mεright) 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 Oleft(mnlogfrac1εright) via accelerated random coordinate descent. This significantly improves over Oleft(fracm√mnDeltaεright) time of the FISTA-based algorithm by Harb et al. (2022). In the high precision regime εllfrac1n where we can even recover the exact solution, our algorithm has a total runtime of Oleft(mnlog nright), 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.