  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 observe
that the former algorithm results in shorter computational times on average,
but also that the latter can outperform the former for specific instance
settings. Finally, 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.
