We investigate a network design problem under traffic uncertainty which arises when provisioning Virtual Private Networks (VPNs): given a set of terminals that must communicate with one another, and a set of possible traffic matrices, sufficient capacity has to be reserved on the links of the large underlying public network so as to support all possible traffic matrices while minimizing the total reservation cost. The problem admits several variants depending on the desired topology of the reserved links, and the nature of the traffic data uncertainty. We present compact linear mixed-integer programming formulations for the problem with the classical hose traffic model and for a new, less conservative, robust variant relying on the traffic statistics that are often available. These flow-based formulations allow to solve optimally medium-to-large-size instances with commercial MIP solvers. We also propose a combined branch-and-price and cutting plane algorithm to tackle larger instances. Computational results obtained for several classes of instances are reported and discussed.
Technical Report n. 16, DEI, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy, January 2005.