  We introduce a class of budgeted prize-collecting covering subgraph problems.
For an input graph with prizes on the vertices and costs on the edges, the aim
of these problems is to find a connected subgraph such that the cost of its
edges does not exceed a given budget and its collected prize is maximum. A
vertex prize is collected when the vertex is visited, but the price can also be
partially collected if the vertex is covered, where an unvisited vertex is
covered by a visited one if the latter belongs to the former's neighbourhood. A
capacity limit is imposed on the number of vertices that can be covered by the
same visited vertex. Potential application areas include network design and
intermodal transportation. We develop a branch-and-cut framework and a Benders
decomposition for the exact solution of the problems in this class. We validate
our algorithmic frameworks for the cases where the subgraph is a tour and a
tree, and for these two cases we also identify novel symmetry-breaking
inequalities.
