Efficient and effective algorithms for densest subgraph discovery and maintenance

Abstract

The densest subgraph problem (DSP) is of great significance due to its wide applications in different domains. Meanwhile, diverse requirements in various applications lead to different density variants for DSP. Unfortunately, existing DSP algorithms cannot be easily extended to handle those variants efficiently and accurately. To fill this gap, we first unify different density metrics into a generalized density definition. We further propose a new model, c-core, to locate the general densest subgraph and show its advantage in accelerating the search process. Extensive experiments show that our c-core-based optimization can provide up to three orders of magnitude speedup over baselines. Methods for maintenance of c-core location are designed to accelerate updates on dynamic graphs. Moreover, we study an important variant of DSP under a size constraint, namely the densest-at-least-k-subgraph (DalkS) problem. We propose an algorithm based on graph decomposition, and it is likely to give a solution that is at least 0.8 of the optimal density in our experiments, while the state-of-the-art method can only ensure a solution with a density of at least 0.5 of the optimal density. Our experiments show that our DalkS algorithm can achieve at least 0.99 of the optimal density for over one-third of all possible size constraints. In addition, we develop an approximation algorithm for the DalkS problem that can be more efficient than the state-of-the-art algorithm while keeping the same approximation ratio of \(\frac{1}{3}\).