Given a directed graph G, the directed densest subgraph (DDS) problem refers to the finding of a subgraph from G, whose density is the highest among all the subgraphs of G. The DDS problem is fundamental to a wide range of applications, such as fraud detection, community mining, and graph compression. However, existing DDS solutions suffer from efficiency and scalability problems: on a threethousand- edge graph, it takes three days for one of the best exact algorithms to complete. In this paper, we develop an efficient and scalable DDS solution. We introduce the notion of [x, y]-core, which is a dense subgraph for G, and show that the densest subgraph can be accurately located through the [x, y]-core with theoretical guarantees. Based on the [x, y]-core, we develop both exact and approximation algorithms. We have performed an extensive evaluation of our approaches on eight real large datasets. The results show that our proposed solutions are up to six orders of magnitude faster than the state-of-the-art.

中文翻译：

---

高效的有向密集子图发现

给定一个有向图 G，有向密集子图 (DDS) 问题是指从 G 中找到一个子图，其密度是 G 的所有子图中最高的。DDS 问题是广泛应用的基础，例如欺诈检测、社区挖掘和图形压缩。然而，现有的 DDS 解决方案存在效率和可扩展性问题：在三千边图上，最好的精确算法之一需要三天才能完成。在本文中，我们开发了一种高效且可扩展的 DDS 解决方案。我们引入了 [x, y]-core 的概念，它是 G 的稠密子图，并表明通过 [x,y]-core 可以准确定位最稠密的子图，并有理论保证。基于 [x, y] 核，我们开发了精确算法和近似算法。我们在八个真实的大型数据集上对我们的方法进行了广泛的评估。结果表明，我们提出的解决方案比最先进的解决方案快六个数量级。