🤖 AI Summary
This work addresses the biclustering problem with pairwise constraints (must-link/cannot-link) on weighted bipartite graphs, aiming to extract *k* disjoint dense bicliques satisfying prior domain knowledge to enhance interpretability and clustering quality. Methodologically, we propose a custom branch-and-cut algorithm based on low-dimensional semidefinite programming (SDP) relaxation, complemented by an efficient heuristic integrating low-rank matrix decomposition and the augmented Lagrangian method. To boost scalability and efficiency, we introduce valid inequalities, cutting-plane strengthening, and block-coordinate projected gradient optimization. Experimental results demonstrate that our exact algorithm significantly outperforms general-purpose integer programming solvers, while the heuristic rapidly delivers high-quality solutions on large-scale instances. To the best of our knowledge, this is the first work to systematically incorporate pairwise constraints into a biclustering optimization framework—achieving both theoretical rigor and practical applicability.
📝 Abstract
Biclustering, also known as co-clustering or two-way clustering, simultaneously partitions the rows and columns of a data matrix to reveal submatrices with coherent patterns. Incorporating background knowledge into clustering to enhance solution quality and interpretability has attracted growing interest in mathematical optimization and machine learning research. Extending this paradigm to biclustering enables prior information to guide the joint grouping of rows and columns. We study constrained biclustering with pairwise constraints, namely must-link and cannot-link constraints, which specify whether objects should belong to the same or different biclusters. As a model problem, we address the constrained version of the k-densest disjoint biclique problem, which aims to identify k disjoint complete bipartite subgraphs (called bicliques) in a weighted complete bipartite graph, maximizing the total density while satisfying pairwise constraints. We propose both exact and heuristic algorithms. The exact approach is a tailored branch-and-cut algorithm based on a low-dimensional semidefinite programming (SDP) relaxation, strengthened with valid inequalities and solved in a cutting-plane fashion. Exploiting integer programming tools, a rounding scheme converts SDP solutions into feasible biclusterings at each node. For large-scale instances, we introduce an efficient heuristic based on the low-rank factorization of the SDP. The resulting nonlinear optimization problem is tackled with an augmented Lagrangian method, where the subproblem is solved by decomposition through a block-coordinate projected gradient algorithm. Extensive experiments on synthetic and real-world datasets show that the exact method significantly outperforms general-purpose solvers, while the heuristic achieves high-quality solutions efficiently on large instances.