Multiplicative Weights Update, Area Convexity and Random Coordinate Descent for Densest Subgraph Problems
Ta Duy Nguyen, Alina Ene
TL;DR
This paper advances the design of scalable algorithms for the densest subgraph problem (DSG) and its decomposition variant by developing three complementary iterative frameworks. The first uses multiplicative weights update (MWU) to solve a width-reduced dual, enabling a simple, binary-search-free reconstruction of the primal densest subgraph with $O\Big(\frac{\log m}{\epsilon^{2}}\Big)$ iterations and near-linear per-iteration time; the second employs area convexity to achieve $O\Big(\frac{\log m}{\epsilon}\Big)$ iterations with a similar time per iteration, improving upon previous results by a factor of the maximum degree; the third approach applies accelerated random coordinate descent to the dense-subgraph decomposition problem, achieving a linear-convergence rate with runtime $O(mn\log\frac{1}{\epsilon})$ and, in high-precision regimes, $O(mn\log n)$ total time. Collectively, these methods provide scalable, parallelizable algorithms with strong theoretical guarantees and practical performance for DSG and its decompositions, matching or surpassing flow-based approaches in many settings. The paper also demonstrates practical efficiency on large graphs and outlines potential extensions to streaming, distributed, and differential privacy contexts.
Abstract
We study the densest subgraph problem and give algorithms via multiplicative weights update and area convexity that converge in $O\left(\frac{\log m}{ε^{2}}\right)$ and $O\left(\frac{\log m}ε\right)$ 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 $Δ$ -- 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 $O\left(mn\log\frac{1}ε\right)$ via accelerated random coordinate descent. This significantly improves over $O\left(\frac{m\sqrt{mnΔ}}ε\right)$ time of the FISTA-based algorithm by Harb et al. (2022). In the high precision regime $ε\ll\frac{1}{n}$ 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.