Summary: The minimum connectivity inference (MCI) problem represents an \(\mathcal{NP}\)-hard generalization of the well-known minimum spanning tree problem. Given a set of vertices and a finite collection of subsets (of this vertex set), the MCI problem requires to find an edge set of minimal cardinality so that the vertices of each subset are connected. Although the problem under consideration has appeared in different application-oriented scientific contexts in the last decades, (efficient) approaches to its exact solution have hardly been addressed in the literature. Currently, even the most promising ILP formulation (an improved flow-based model) can only deal with rather small instances in reasonable times. In order to also tackle practically relevant problem sizes, our contribution is twofold: At first, we propose several new modeling frameworks for the MCI problem and investigate their theoretical properties as well as their computational behavior. Moreover, we introduce the concepts of simple model reduction and generalized model reduction which can be applied to reduce the numbers of variables and/or constraints in the various formulations. Based on extensive numerical experiments, the practical advantages of these principles are validated.