A basic question in Virtual Private Network (VPN) design is if the symmetric version of the problem always has an optimal solution which is a tree network. An affirmative answer would imply that the symmetric VPN problem is solvable in polynomial time. We give an affirmative answer in case the communication network within which to create the VPN is a circuit. This seems to be an important step towards an answer to the general question. The proof relies on a dual pair of linear programs and actually implies an even stronger property of VPNs. We show that this property also holds for some other special cases of the problem.
Springer
Lecture Notes in Computer Science
International Conference on Integer Programming and Combinatorial Optimization
Networks and Optimization

Hurkens, C., Keijsper, J. C. M., & Stougie, L. (2005). Virtual private network design: a proof of the tree routing conjecture on ring networks. In Proceedings of the 11th International Conference on Integer Programming and Combinatorial Optimization (pp. 407–421). Springer.