Algorithms for Densest Subgraphs of Vertex-Weighted Graphs

Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem...

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Bibliographic Details
Published inMathematics (Basel) Vol. 12; no. 14; p. 2206
Main Authors Liu, Zhongling, Chen, Wenbin, Li, Fufang, Qi, Ke, Wang, Jianxiong
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.07.2024
Subjects
Algorithms
Approximation
Computer vision
densest subgraph
Density
graph mining
Graph theory
Graphs
Linear programming
Polynomials
Social networks
Subgroups
vertex-weighted graph
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Summary:Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding a subgraph with maximum mean density. However, most densest subgraph-finding algorithms are based on edge-weighted graphs, where edge weights can only represent a single value dimension, whereas practical applications involve multiple dimensions. To resolve the challenge, we propose two algorithms for resolving the densest subgraph problem in a vertex-weighted graph. First, we present an exact algorithm that builds upon Goldberg’s original algorithm. Through theoretical exploration and analysis, we rigorously verify our proposed algorithm’s correctness and confirm that it can efficiently run in polynomial time O(n(n + m)log2n) is its temporal complexity. Our approach can be applied to identify closely related subgroups demonstrating the maximum average density in real-life situations. Additionally, we consistently offer an approximation algorithm that guarantees an accurate approximation ratio of 2. In conclusion, our contributions enrich theoretical foundations for addressing the densest subgraph problem.
ISSN:2227-7390
2227-7390
DOI:10.3390/math12142206