Revisiting Maximum $k$-Biplex Search Through $k$-Bounded-Degree Deletion

2026-07-08

AI Summary

The maximum $k$-biplex problem plays a crucial role in community detection, recommender systems, and fraud detection, yet its NP-hard nature has hindered the development of efficient exact algorithms. This work addresses this challenge by introducing a novel algorithmic framework rooted in the complement graph perspective, which uncovers a structural duality between the maximum $k$-biplex problem and the minimum $k$-bounded-degree deletion problem. The proposed approach integrates upper-bound pruning with a high-quality initial-solution heuristic. Theoretical analysis yields, for the first time, a worst-case time complexity strictly below $O^*(2^n)$. Extensive experiments on eight real-world bipartite graphs demonstrate that the method achieves up to four orders of magnitude speedup over the current state-of-the-art exact algorithm and enables efficient computation even for relatively large values of $k$, such as $k=3$.

Abstract

Biplex, as a relaxation of the biclique model, has emerged as an important cohesive subgraph model for bipartite graph analysis. The maximum $k$-biplex search problem aims to identify the $k$-biplex with maximum number of edges and has been widely applied in various real-world applications, including community detection, online recommendation, and fraud detection. However, the problem is NP-hard, and existing exact algorithms remain inefficient on large-scale bipartite graphs with large values of $k$ (e.g., $k\geq 3$). In this paper, we revisit the maximum $k$-biplex search problem from a complementary perspective. We reveal a novel structural duality: finding a maximum $k$-biplex in a bipartite graph is equivalent to finding a minimal $k$-bounded-degree deletion in its complement graph. Based on this observation, we propose a novel deletion-based algorithm for the maximum $k$-biplex search problem. We theoretically prove that the proposed algorithm achieves a worst-case time complexity of $O^*(γ_k^n)$, where $γ_k<2$. Specifically, $γ_1=1.725$, $γ_2=1.856$, and $γ_3=1.928$. To further enhance practical efficiency, we develop several effective upper-bounding techniques and a heuristic strategy for obtaining high-quality initial solutions, which substantially reduce the search space. Extensive experiments on eight real-world bipartite graphs demonstrate the efficiency of our approach, which achieves up to four orders of magnitude speedups over state-of-the-art algorithms.

Problem

• maximum k-biplex search
• bipartite graph
• NP-hard
• k-bounded-degree deletion
• cohesive subgraph

Innovation

• k-biplex
• bounded-degree deletion
• structural duality
• exact algorithm
• bipartite graph