The minimum sum-of-squares clustering problem (MSSC), also known as k-means clustering, refers to the problem of partitioning n data points into k clusters, with the objective of minimizing the total sum of squared Euclidean distances between each point and the center of its assigned cluster. We propose an efficient algorithm for solving large-scale MSSC instances, which combines column generation (CG) with dynamic constraint aggregation (DCA) to effectively reduce the number of constraints considered in the CG master problem. DCA was originally conceived to reduce degeneracy in set partitioning problems by utilizing an aggregated restricted master problem obtained from a partition of the set partitioning constraints into disjoint clusters. In this work, we explore the use of DCA within a CG algorithm for MSSC exact solution. Our method is fine-tuned by a series of ablation studies on DCA design choices, and is demonstrated to significantly outperform existing state-of-the-art exact approaches available in the literature. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada [Grant 2023-04466]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0938 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0938 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Biclustering, also called co-clustering, block clustering, or two-way clustering, involves the simultaneous clustering of both the rows and the columns of a data matrix into distinct groups such that the rows and columns within a group display similar patterns. As a model problem for biclustering, we consider the k-densest disjoint biclique problem, whose goal is to identify k disjoint complete bipartite subgraphs (called bicliques) of a given weighted complete bipartite graph such that the sum of their densities is maximized. To address this problem, we present a tailored branch-and-cut algorithm. For the upper-bound routine, we consider a semidefinite programming relaxation and propose valid inequalities to strengthen the bound. We solve this relaxation in a cutting-plane fashion using a first-order method. For the lower bound, we design a maximum weight matching rounding procedure that exploits the solution of the relaxation solved at each node. Computational results on both synthetic and real-world instances show that the proposed algorithm can solve instances approximately 20 times larger than those handled by general-purpose solvers. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0683 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0683 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

The minimum sum-of-squares clustering problem (MSSC) consists of partitioning n observations into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure in which valid inequalities are iteratively added to the Peng–Wei semidefinite programming (SDP) relaxation. The upper bound is computed with the constrained version of k-means in which the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level, preserving the structure of the SDP problem itself. To the best of our knowledge, the obtained results show that the approach allows us to successfully solve, for the first time, real-world instances up to 4,000 data points.