Densest Subgraph: Supermodularity, Iterative Peeling, and Flow

The densest subgraph problem in a graph (DSG), in the simplest form, is the following. Given an undirected graph G = (V, E) find a subset S ⊆ V of vertices that maximizes the ratio |E(S)|/|S| where E(S) is the set of edges with both endpoints in S. DSG and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. In this paper we study fast algorithms and structural aspects of DSG via the lens of supermodularity. For this we consider the densest supermodular subset problem (DSS): given a non-negative supermodular function f : 2V → ℝ+, maximize f(S)/|S|. For DSG we describe a simple flow-based algorithm that outputs a (1–∊)-approximation in deterministic Õ(m/∊) time where m is the number of edges. Our algorithm is the first to have a near-linear dependence on m and 1/∊ and improves previous methods based on an LP relaxation. It generalizes to hypergraphs, and also yields a faster algorithm for directed DSG. Greedy peeling algorithms have been very popular for DSG and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for DSS and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al. [12] developed an iterative peeling algorithm for DSG which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges to a (1–∊)-approximation for any supermodular function f; the key to our proof is to consider an LP formulation that is derived via the Lovász extension of a supermodular function. For DSG the bound on the number of iterations we prove is where Δ is the maximum degree and λ∗ is the optimum value. Our work suggests that iterative peeling can be an effective heuristic for several objectives considered in the literature. Finally, we show that the 2-approximation for densest-at-least-k subgraph [37] extends to the supermodular setting. We also give a unified analysis of the peeling algorithm for this problem, and via this analysis derive an approximation guarantee for a generalization of DSS to maximize f(S)/g(|S|) for a concave function g.