A Branch-and-Bound Approach for Maximum Low-Diameter Dense Subgraph Problems

2025-11-05

AI Summary

This paper addresses the problem of mining maximum *f*(·)-dense subgraphs with diameter at most two, aiming to enhance both connectivity and cohesiveness of cohesive subgraphs. To tackle the challenge that *f*(·)-dense subgraphs may be disconnected, we propose a graph-decomposition-based branch-and-bound algorithm: (i) decompose the input graph into *n* subgraphs for parallel optimization; (ii) employ a two-hop core ordering to accelerate search; and (iii) design a tight upper-bound estimation method based on vertex ordering, integrated with mixed-integer programming (MIP) modeling and bound-driven pruning. Experiments on 139 real-world graphs demonstrate that our method significantly outperforms standard MIP solvers and conventional branch-and-bound approaches under two representative *f*(·) functions—achieving nearly double the number of optimal solutions within one hour. The results validate its efficiency, scalability, and practical effectiveness for dense subgraph discovery under diameter constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

A graph with $n$ vertices is an $f(cdot)$-dense graph if it has at least $f(n)$ edges, $f(cdot)$ being a well-defined function. The notion $f(cdot)$-dense graph encompasses various clique models like $gamma$-quasi cliques, $k$-defective cliques, and dense cliques, arising in cohesive subgraph extraction applications. However, the $f(cdot)$-dense graph may be disconnected or weakly connected. To conquer this, we study the problem of finding the largest $f(cdot)$-dense subgraph with a diameter of at most two in the paper. Specifically, we present a decomposition-based branch-and-bound algorithm to optimally solve this problem. The key feature of the algorithm is a decomposition framework that breaks the graph into $n$ smaller subgraphs, allowing independent searches in each subgraph. We also introduce decomposition strategies including degeneracy and two-hop degeneracy orderings, alongside a branch-and-bound algorithm with a novel sorting-based upper bound to solve each subproblem. Worst-case complexity for each component is provided. Empirical results on 139 real-world graphs under two $f(cdot)$ functions show our algorithm outperforms the MIP solver and pure branch-and-bound, solving nearly twice as many instances optimally within one hour.

Problem

• Finding largest dense subgraph with diameter at most two
• Solving disconnected or weakly connected dense graph issues
• Developing branch-and-bound algorithm for optimal subgraph extraction

Innovation

• Decomposition framework breaks graph into subgraphs
• Uses degeneracy and two-hop ordering strategies
• Branch-and-bound with sorting-based upper bound