Approximating Some Network Design Problems with Node Costs

We study several multi-criteria undirected network design problems with node costs and lengths with all problems related to the node costs Multicommodity Buy at Bulk (MBB) problem in which we are given a graph G = (V,E), demands {dst : s, t ∈ V }, and a family {cv : v ∈ V } of subadditive cost functions. For every s, t ∈ V we seek to send dst flow units from s to t on a single path, so that