The VPN Problem with Concave Costs

Only recently Goyal, Olver and Shepherd (Proc. STOC, 2008) proved that the symmetric Virtual Private Network Design (sVPN) problem has the tree routing property, namely, that there always exists an optimal solution to the problem whose support is a tree. Combining this with previous results by Fingerhut, Suri and Turner (J. Alg., 1997) and Gupta, Kleinberg, Kumar, Rastogi and Yener (Proc. STOC, 2001), sVPN can be solved in polynomial time. In this paper we investigate a NP-hard generalization of sVPN, where the contribution of each edge to the total cost is proportional to some concave, non-decreasing function of the capacity reservation. We show that the tree routing property extends to the new problem, and give a 24.92approximation algorithm for it. We also show that the undirected uncapacitated single-source minimum concave-cost (fractional) flow problem has always the tree property when the cost function has some property of symmetry.