On the Complexity of the Asymmetric VPN Problem

We give the first constant factor approximation algorithm for the asymmetric Virtual Private Network (Vpn) problem with arbitrary concave costs. We even show the stronger result, that there is always a tree solution of cost at most 2·OPT and that a tree solution of (expected) cost at most 49.84 ·OPT can be determined in polynomial time. For the case of linear cost we obtain a (2+ε S )-approximation algorithm for any fixed ε > 0, where S and R (R ≥ S) denote the outgoing and ingoing demand, respectively. Furthermore, we answer an outstanding open question about the complexity status of the so called balanced Vpn problem by proving its NPhardness.