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A Survey on the Densest Subgraph Problem and Its Variants

Tommaso Lanciano, Atsushi Miyauchi, Adriano Fazzone, Francesco Bonchi

TL;DR

This survey provides a comprehensive account of the Densest Subgraph Problem (DSP) and its many variants. It surveys exact and approximation algorithms, deep connections to max-flow, LP, and submodular minimization, and constrained formulations. It then covers richer graph models (weighted, directed, multilayer, temporal, uncertain, hypergraphs, and metric spaces) and practical settings (streaming, distributed, and real-world applications). The work highlights recent breakthroughs (2022–2023) and enumerates open problems, including convergence of iterative peeling, tighter approximations for constrained variants, and broader applicability to complex graph families. Overall, it positions DSP as a central, evolving tool for extracting dense structures across diverse domains and data models.

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.

A Survey on the Densest Subgraph Problem and Its Variants

TL;DR

This survey provides a comprehensive account of the Densest Subgraph Problem (DSP) and its many variants. It surveys exact and approximation algorithms, deep connections to max-flow, LP, and submodular minimization, and constrained formulations. It then covers richer graph models (weighted, directed, multilayer, temporal, uncertain, hypergraphs, and metric spaces) and practical settings (streaming, distributed, and real-world applications). The work highlights recent breakthroughs (2022–2023) and enumerates open problems, including convergence of iterative peeling, tighter approximations for constrained variants, and broader applicability to complex graph families. Overall, it positions DSP as a central, evolving tool for extracting dense structures across diverse domains and data models.

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.
Paper Structure (31 sections, 6 theorems, 19 equations, 1 figure, 4 algorithms)

This paper contains 31 sections, 6 theorems, 19 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Fundamental Results
  3. Exact algorithms for DSP
  4. Goldberg's maximum-flow-based algorithm
  5. Charikar's LP-based algorithm
  6. An exact algorithm based on submodular function minimization
  7. Approximation algorithms for DSP
  8. Greedy peeling algorithm
  9. Iterative greedy peeling algorithm and beyond
  10. Adding Constraints
  11. Size constraints
  12. Densest $k$-subgraph problem
  13. Densest at-least-$k$-subgraph problem and densest at-most-$k$-subgraph problem
  14. Seed set
  15. Connectivity constraints
  16. ...and 16 more sections

Key Result

Lemma 1

Let $(X,Y)$ be an $s$--$t$ cut of the edge-weighted directed graph $(U,A,w_\beta)$, and $S=X\setminus \{s\}$. Then it holds that $\mathrm{cost}(X,Y)=m+\beta |S|-e[S].$ In particular, when $X=\{s\}$, $\mathrm{cost}(X,Y)=m$ holds.

Figures (1)

  • Figure 1: An edge-weighted directed graph $(U,A,w_\beta)$ constructed from $G$ and $\beta$.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 2 more