🤖 AI Summary
This paper addresses the segmentation optimization problem on edge-colored graphs: partitioning a graph into color-connected components such that each color appears at most once per component. We formulate three variants—minimizing the number of components, maximizing monochromatic isolation, and others—and propose, for the first time, an exact branch-and-cut algorithm based on integer nonlinear programming with standard linearization. Our method introduces novel contributions: (i) custom valid inequalities; (ii) tight variable bounds; (iii) effective preprocessing rules; (iv) a warm-start strategy; and (v) bound-based pruning to accelerate convergence. Evaluated on benchmark instances, the solver efficiently handles medium-scale problems and constitutes the first scalable exact approach for this problem class. It provides a new computational tool for applications in community detection, cybersecurity, and bioinformatics—particularly in analyzing colored structural patterns.
📝 Abstract
We tackle three optimization problems in which a colored graph, where each node is assigned a color, must be partitioned into colorful connected components. A component is defined as colorful if each color appears at most once. The problems differ in the objective function, which determines which partition is the best one. These problems have applications in community detection, cybersecurity, and bioinformatics. We present integer nonlinear formulations, which are then linearized using standard techniques. To solve these formulations, we develop exact branch-and-cut algorithms, embedding various improving techniques, such as valid inequalities, bounds limiting the number of variables, and warm-start and preprocessing techniques. Extensive computational tests on benchmark instances demonstrate the effectiveness of the proposed procedures. The branch-and-cut algorithms can solve reasonably sized instances efficiently. To the best of our knowledge, we are the first to propose an exact algorithm for solving these problems. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms & Applications. Funding: The work of M. Cerulli was partially funded by project “SEcurity and RIghts in the CyberSpace” [Grant PE00000014] under the MUR National Recovery and Resilience Plan funded by the European Union - NextGenerationEU. 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.2024.0927 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0927 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .