On Directed Densest Subgraph Detection

Abstract

The well-studied directed densest subgraph problem aims to find two (possibly overlapping) vertex subsets $S^*$ and $T^*$ in a given directed graph $G=(V,E)$ such that $\rho (S,T) = \frac{|E(S,T)|}{\sqrt{|S||T|}}$ is maximized; here E(S, T) denotes the set of edges from vertices of S to T in G. This problem is polynomial-time solvable, and both exact algorithms and approximation algorithms have been proposed in the literature. However, the existing exact algorithms are time-consuming, while the existing approximation algorithms often yield trivial solutions that consist of the highest-degree vertex and its in-neighbors or out-neighbors. Moreover, there is nothing special about geometric mean that is adopted in the existing density measure for combining $\frac{|E(S,T)|}{|S|}$ and $\frac{|E(S,T)|}{|T|}$. In this paper, we explore alternative density measures and propose corresponding algorithms, for directed densest subgraph identification. Specifically, we introduce three density measures that combine $\frac{|E(S,T)|}{|S|}$ and $\frac{|E(S,T)|}{|T|}$ by harmonic mean, arithmetic mean, and minimum mean, respectively. Based on these density measures, we formulate the harmonic mean-based directed densest subgraph (HDDS) problem, the arithmetic mean-based directed densest subgraph (ADDS) problem, and the minimum mean-based directed densest subgraph (MDDS) problem. We then propose a 2-approximation algorithm for HDDS, a 2-approximation algorithm for ADDS, and a heuristic algorithm for MDDS; our HDDS and MDDS algorithms run in linear time to the input graph size. Extensive empirical studies on large real-world directed graphs show that our ADDS algorithm produces similar trivial results as the existing approximation algorithm, and our HDDS and MDDS algorithms generate nontrivial and much better solutions and scale to large graphs.