Pricing tree access networks with connected backbones

Consider the following network subscription pricing problem. We are given a graph G= (V, E) with a root r, and potential customers are companies headquartered at r with locations at a subset of nodes. Every customer requires a network connecting its locations to r. The network provider can build this network with a combination of backbone edges (consisting of high capacity cables) that can route any subset of the customers, and access edges that can route only a single customer's traffic. The backbone edges cost M times that of the access edges. Our goal is to devise a group-strategyproof pricing mechanism for the network provider, i.e., one in which truth-telling is the optimal strategy for the customers, even in the presence of coalitions. We give a pricing mechanism that is 2-competitive and O(1)-budget-balanced. As a means to obtaining this pricing mechanism, we present the first primaldual 8-approximation algorithm for this problem. Since the two-stage Stochastic Steiner tree problem can be reduced to the underlying network design, we get a primal-dual algorithm for the stochastic problem as well. Finally, as a byproduct of our techniques, we also provide bounds on the inefficiency of our mechanism.