Densest subgraph discovery (DSD) is a fundamental topic in graph mining. It has been studied for decades, and is widely used in various areas, including network science, biological analysis, and graph databases. As a typical problem of DSD, the k -clique densest subgraph (CDS) problem aims to detect a subgraph from a graph, such that the number of k -cliques over the number of its vertices is maximized. While the CDS problem has received plenty of attention in the literature, existing CDS algorithms that perform best in practice often have weaker theoretical guarantees, while those with the stronger theoretical assurances tend to perform worse in practice. Besides, all the existing CDS algorithms struggle with graphs with high degeneracy values, a characteristic commonly found in real-world graphs. To bridge the huge gap between practice and theory, in this paper, we first introduce a novel graph reduction technique, which locates the CDS into a very small subgraph, with non-trivial theoretical guarantees. We further propose a new efficient approximation algorithm by employing the state-of-the-art k -clique counting algorithm, which shares all the advantages of existing algorithms, achieving both strong practical efficiency and theoretical guarantees. Extensive experiments on 12 real-world large graphs demonstrate the high efficiency of our CDS algorithm. Particularly, our algorithm is up to four orders of magnitude faster than the state-of-the-art algorithm while maintaining the same accuracy guarantees and requiring much less memory.
Zhou et al. (Sun,) studied this question.