We initiate the theoretical study of the problem of minimizing the size of an iBGP (Interior Border Gateway Protocol) overlay in an Autonomous System (AS) in the Internet subject to a natural notion of correctness derived from the standard "hot-potato" routing rules. For both natural versions of the problem (where we measure the size of an overlay by either the number of edges or the maximum degree) we prove that it is NP-hard to approximate to a factor better than Ω(log n) and provide approximation algorithms with ratio O(n~(1/2)). This algorithm is based on a natural LP relaxation and randomized rounding technique inspired by recent progress on approximating directed spanners. The main technique we use is a reduction to a new connectivity-based network design problem that we call Constrained Connectivity, in which we are given a graph G = (V, E) and for every pair of vertices u, v € V we are given a set S(u, v) is contained in V called the safe set of the pair. The goal is to find the smallest subgraph H = (V, F) of G in which every pair of vertices u, v is connected by a path contained in S(u, v). We show that the iBGP problem can be reduced to the special case of Constrained Connectivity where G = K_n. Furthermore, we believe that Constrained Connectivity is an interesting problem in its own right, so provide stronger hardness results and integrality gaps for the general case.
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