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

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.