  Capacitated fixed-charge network flows are used to model a variety of
problems in telecommunication, facility location, production planning and
supply chain management. In this paper, we investigate capacitated path
substructures and derive strong and easy-to-compute \emph{path cover and path
pack inequalities}. These inequalities are based on an explicit
characterization of the submodular inequalities through a fast computation of
parametric minimum cuts on a path, and they generalize the well-known flow
cover and flow pack inequalities for the single-node relaxations of
fixed-charge flow models. We provide necessary and sufficient facet conditions.
Computational results demonstrate the effectiveness of the inequalities when
used as cuts in a branch-and-cut algorithm.
