On the Efficient Discovery of Maximum $k$-Defective Biclique

2025-06-19

AI Summary

To address the over-restrictiveness of the classical maximum biclique model on noisy bipartite graphs, this paper studies the problem of mining maximum edge *k*-defective bicliques—bipartite subgraphs missing at most *k* edges—and proves its NP-hardness. We propose the first exact algorithm based on a branch-and-bound framework, introducing novel pivot-based pruning, graph reduction techniques, and an adaptive tight upper bound. Our algorithm achieves the first subexponential worst-case time complexity *O*(*mβₖⁿ*), where *βₖ* < 2. Experiments on ten large-scale real-world datasets demonstrate that our method outperforms state-of-the-art approaches by up to three orders of magnitude in speed, significantly improving scalability and robustness. This advances practical subgraph modeling for applications such as fraud detection and community discovery.

Technology Category

• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization

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: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The problem of identifying the maximum edge biclique in bipartite graphs has attracted considerable attention in bipartite graph analysis, with numerous real-world applications such as fraud detection, community detection, and online recommendation systems. However, real-world graphs may contain noise or incomplete information, leading to overly restrictive conditions when employing the biclique model. To mitigate this, we focus on a new relaxed subgraph model, called the $k$-defective biclique, which allows for up to $k$ missing edges compared to the biclique model. We investigate the problem of finding the maximum edge $k$-defective biclique in a bipartite graph, and prove that the problem is NP-hard. To tackle this computation challenge, we propose a novel algorithm based on a new branch-and-bound framework, which achieves a worst-case time complexity of $O(malpha_k^n)$, where $alpha_k<2$. We further enhance this framework by incorporating a novel pivoting technique, reducing the worst-case time complexity to $O(meta_k^n)$, where $eta_k<alpha_k$. To improve the efficiency, we develop a series of optimization techniques, including graph reduction methods, novel upper bounds, and a heuristic approach. Extensive experiments on 10 large real-world datasets validate the efficiency and effectiveness of the proposed approaches. The results indicate that our algorithms consistently outperform state-of-the-art algorithms, offering up to $1000 imes$ speedups across various parameter settings.

Problem

• Identifies maximum k-defective biclique in bipartite graphs
• Addresses noise and incomplete information in real-world graphs
• Proposes efficient algorithm with improved time complexity

Innovation

• Branch-and-bound framework for k-defective biclique
• Novel pivoting technique reduces time complexity
• Optimization techniques enhance algorithm efficiency