Virtual private network design over the first Chvátal closure

In this paper we consider the virtual private network (VPN) design problem. Given upper bounds on the amount of traffic that an endpoint could send or receive, the problem needs to reserve enough capacities in such a way that any demand matrix that respects the upper bound could be routed without exceeding the reserved capacities and the total reservation cost is minimized. In On the difficulty of virtual private network instances (Networks 63 (2014) 327-333), we argued that the computational investigation on exact mathematical programming approaches for VPN needs to be revised after that challenging instances have been exposed. To that end, we consider the VPN design problem over the first Chvatal closure and demonstrate that tight solutions could be found for the VPN design problem only by optimizing over the closure. First, we perform theoretical investigation on adding rank-1 Chvatal-Gomory cuts to the problem. Along the way, an important property for such cuts is proved that omits a large number of redundant rank-1 cuts. We then provide interesting insights about the problem and reduce the existing MIP formulations to a binary one. On the computational side, we investigate the idea of adding rank-1 cuts more aggressively in order to computationally evaluate tightness of the first Chvatal closure for the VPN design problem. Here, the binary reduction plays an important role allowing the use of special cuts of the first closure, namely the zero-half cuts. We show that, almost all the success of the first Chvatal closure of the VPN design problem in raising dual bound is due to zero-half cuts. Our experiments on the benchmark instances in this article show that a state-of-the-art IP solver without using zero-half cuts could not even hit the challenging benchmarks. As a results a cut-and-branch framework that aggressively adds such cuts at the root could solve the challenging VPN instances to the extent of zero or small integrality gap in a reasonable amount of time.