Compact Models for Some Cluster Problems on Node-Colored Graphs

Three optimization problems based on node-colored undirected graphs are the subject of the present study. These problems model real-world applications in several domains, such as cybersecurity, bioinformatics, and social networks, although they have a similar abstract representation. In all of the problems, the goal is to partition the graph into colorful connected components, which means that in each of the connected components, a color can appear in at most one node. The problems are optimized according to different objective functions, leading to different optimal partitions. We propose a compact Mixed Integer Linear Programming formulation for each of the three problems. These models are based on spanning trees, represented through multi-commodity flows. The compact nature of the new linear models is easier to handle than the approaches that previously appeared in the literature. These were based on models with an exponential number of constraints, which, therefore, required complex solving techniques based on the dynamic generation of constraints within a branch-and-cut framework. Computational experiments carried out on the standard benchmark instances for the problems show the potential of the new compact methods, which, once fed into modern state-of-the-art solvers, are able to obtain results better than the previous algorithmic approaches. As an outcome of the experimental campaign, a dozen instances of the different problems considered are closed for the first time.