A Survey on the Densest Subgraph Problem and Its Variants

TL;DR

This survey comprehensively reviews the Densest Subgraph Problem, its variants, recent breakthroughs, and applications, highlighting ongoing research challenges and future directions in this well-studied area.

Contribution

It provides an exhaustive overview of fundamental results, recent advances, and open problems related to the Densest Subgraph Problem and its variants.

Findings

Recent breakthroughs in 2022-2023 significantly advanced understanding.

The survey covers a wide range of variants and applications.

Open problems remain in algorithm efficiency and problem generalizations.

Abstract

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 since the early 1970s, with many variants proposed and many applications built on top of this basic definition. Recent years have witnessed a revival of research interest in this problem with several important 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 to the most recent results. The survey also presents a comprehensive overview of applications and discusses some interesting open problems for this evergreen research topic.