The capacity formulation of the capacitated edge activation problem

Given a capacitated network, the Capacitated Edge Activation problem consists of choosing the edges to be activated to ensure the routing of a set of demands. We focus on capacity formulations of the problem, that is, formulations including only design variables, whereas variables corresponding to the routes are projected out. First, we investigate the combinatorial properties of the problem to derive a capacity formulation both for splittable flows (the routes are unrestricted) and for unsplittable ones (each demand must be routed on a single path). Then, we study the corresponding polyhedron, identifying valid and facet-defining inequalities. Finally, we develop a branch-and-cut algorithm and present computational results.