Several important tasks in medical image analysis can be stated in the form of an optimization problem whose feasible solutions are connected subgraphs. Examples include the reconstruction of neural or vascular structures under connectedness constraints. We discuss the minimum cost connected subgraph (MCCS) problem and its approximations from the perspective of medical applications. We propose a) objective-dependent constraints and b) novel constraint generation schemes to solve this optimization problem exactly by means of a branch-and-cut algorithm. These are shown to improve scalability and allow us to solve instances of two medical benchmark datasets to optimality for the first time. This enables us to perform a quantitative comparison between exact and approximative algorithms, where we identify the geodesic tree algorithm as an excellent alternative to exact inference on the examined datasets.