The capacity formulation of the capacitated edge activation problem

Abstract

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.

An optimization approach for congestion control in network routing with quality of service requirements

“…In , among the others, the problem of choosing minimum cost integer capacities for the arcs of a network to ensure the routing of a set of commodities has been studied and different solution approaches and formulations have been proposed. In , instead, arc capacities have already been set and the decision is to choose the arcs to be activated to ensure the routing of some traffic demands. Two‐layer networks are investigated in and references therein.…”

A polyhedral analysis of the capacitated edge activation problem with uncertain demands

“…Hence, some of the edges are usually deactivated and switched on only when something unusual (e.g., a fault) happens . The capacitated edge activation (CEA) problem selects the minimum cost set of edges to be activated to guarantee the routing of the demands . Differently from the NL problem, where integer capacities must be computed and no upper bound restrictions on them are present, the CEA problem has fixed and bounded capacities.…”

“…For splittable flows, capacity formulations can be obtained by the Benders decomposition approach , a popular technique with applications to many problems . Benders approaches and capacity formulations for network design problems with and without uncertainty can be found in and references therein. For unsplittable flows, capacity formulations are far less common and they are mainly obtained using (possibly modified versions of) combinatorial Benders cuts derived from the combinatorial properties, if any, of the considered problem .…”

“…Since CEA has bounded capacities, an instance of CEA cannot be transformed into an instance of NL. Capacity formulations for the CEA problem and related polyhedral results have been investigated in . Valid inequalities and polyhedral results for the capacity formulation of the CEA problem with splittable and unsplittable flows are presented in .…”

“…Capacity formulations for the CEA problem and related polyhedral results have been investigated in . Valid inequalities and polyhedral results for the capacity formulation of the CEA problem with splittable and unsplittable flows are presented in . The author shows that the tight metric inequalities, that completely define the polyhedron of the capacity formulation of many versions of the NL problem , do not provide the complete description of the polyhedron corresponding to the CEA problem.…”