A Survey on the Densest Subgraph Problem and its Variants

The Densest Subgraph Problem requires to find, in a given graph, a subset of vertices whose induced subgraph maximizes a measure of density. The problem has received a great deal of attention in the algorithmic literature over the last five decades, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest on this problem with several interesting contributions, including some groundbreaking results, published in 2022 and 2023. This survey provides a deep overview of the fundamental results and an exhaustive coverage of the many variants proposed in the literature, with a special attention on the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic.