Approximating minimum cost connectivity problems

We survey approximation algorithms and hardness results for versions of the Generalized Steiner Network (GSN) problem in which we seek to find a low cost subgraph (where the cost of a subgraph is the sum of the costs of its edges) that satisfies prescribed connectivity requirements. These problems include the following well known problems: min-cost k-flow, min-cost spanning tree, traveling salesman, directed/undirected Steiner Tree, Steiner forest, k-edge/node-connected spanning subgraph, and others. The type of problems we consider can be formally defined using the following unified framework. Let G = (V,E) be a (possibly directed) graph and let S ⊆ V . The S-connectivity λG(u, v) of (u, v) in G is the maximum number of uv-paths such that no two of them have an edge or a node in S − {u, v} in common. Generalized Steiner Network (GSN) Instance: A (possibly directed) graph G = (V, E) with costs {ce : e ∈ E} on the edges, a node subset S ⊆ V , and a nonnegative integer requirement function r(u, v) on V × V . Objective: Find a minimum cost spanning (that is, on the same node set) subgraph G = (V,E) of G so that