Flowless: Extracting Densest Subgraphs Without Flow Computations

The problem of finding dense components of a graph is a major primitive in graph mining and data analysis. The densest subgraph problem (DSP) that asks to find a subgraph with maximum average degree forms a basic primitive in dense subgraph discovery with applications ranging from community detection to unsupervised discovery of biological network modules [16]. The DSP is exactly solvable in polynomial time using maximum flows [14, 17, 22]. Due to the high computational cost of maximum flows, Charikar's greedy approximation algorithm is usually preferred in practice due to its linear time and linear space complexity [3, 8]. It constitutes a key algorithmic idea in scalable solutions for large-scale dynamic graphs [5, 7]. However, its output density can be a factor 2 off the optimal solution.In this paper we design GREEDY++, an iterative peeling algorithm that improves upon the performance of Charikar's greedy algorithm significantly. Our iterative greedy algorithm is able to output near-optimal and optimal solutions fast by adding a few more passes to Charikar's greedy algorithm. Furthermore GREEDY++ is more robust against the structural heterogeneities (e.g., skewed degree distributions) in real-world datasets. An additional property of our algorithm is that it is able to assess quickly, without computing maximum flows, whether Charikar's approximation quality on a given graph instance is closer to the worst case theoretical guarantee of 1/2 or to optimality. We also demonstrate that our method has significant efficiency advantage over the maximum flow based exact optimal algorithm. For example, our algorithm achieves similar to 145x speedup on average across a variety of real-world graphs while finding subgraphs of density that are at least 90% as dense as the densest subgraph.