We consider the following network design problem. We are given an undirected graph G = (V, E) with edges costs c(e) and a set of terminal nodes W. A hose matrix for W is any symmetric matrix (Dij) such that for each i, � j�=i Dij ≤ 1. We must compute the minimum cost edge capacities that are able to support the oblivious routing of every hose matrix in the network. An oblivious routing template, in this context, is a simple path Pij for each pair i, j ∈ W. Given such a template, if we are to route a demand matrix D, then for each i, j we send Dij units of flow along each Pij. Fingerhut et al.  and Gupta et al.  obtained a 2-approximation for this problem, using a routing template in the form of a tree. It has been widely asked (cf. ) if this solution actually results in the optimal capacity for the single path VPN design problem; this has become known as the VPN conjecture. The conjecture has previously been proven for some restricted classes of graphs [18, 15, 14]. Our main theorem establishes that this conjecture is true in general graphs. This also settles the complexity of the single path VPN problem. We also show that the version of the conjecture where fractional routing templates are allowed is false
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