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

TL;DR

This paper introduces a new algorithmic framework for efficiently discovering maximum $k$-defective bicliques in bipartite graphs, addressing noise and incompleteness in real-world data with significant speed improvements.

Contribution

It proposes a novel branch-and-bound algorithm with pivoting and optimization techniques for finding maximum $k$-defective bicliques, improving computational efficiency and scalability.

Findings

Achieves up to 1000x speedup over existing methods

Validates effectiveness on 10 large real-world datasets

Demonstrates the algorithm's scalability and robustness

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 $O(m\alpha_k^n)$, where $\alpha_k < 2$. We further enhance this framework by incorporating a novel pivoting technique, reducing the worst-case time complexity to $O(m\beta_k^n)$, where $\beta_k < \alpha_k$. 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 $1000\times$ speedups across various parameter settings.