The asymmetric VPN tree problem: polyhedral results and Branch-and-Cut

In this paper we consider a variant of the virtual private network design problem (VPNDP). Given an uncapacitated physical network, represented by a graph G=(V∪P,E), where V is the set of VPN routers and P is the set of clients for which it is given thresholds on the amount of traffic that each client can send (bp+) or receive (bp−), the VPNDP asks for (1) a connected sub-network G′=(V′∪P,E′), (2) a client assignments (p, v), p∈P and v∈V′, and (3) a bandwidth allocation ue,e∈E′ in order to accommodate any traffic demand matrix that respects client thresholds. When G′ is acyclic, we have a VPN tree (VPNT). Also, when client thresholds are asymmetric, i.e., ∑p∈Pbp+≠∑b∈Pbp−, the problem has been shown to be NP-hard. In this paper, we give MILP formulations for the asymmetric VPN tree problem. Also, we discuss the polytope associated with one of these formulations and describe several classes of valid inequalities. Moreover, we present necessary and sufficient conditions under which these inequalities define facets. We also devise separation routines. Using these routines, we propose a Branch-and-Cut algorithm and present a computational study.