Two-level hub Steiner trees

Abstract

We study a fundamental class of two-layer network design problems. A hub layer is configured by establishing hubs at selected nodes at considerable cost so that the routes between hubs can be operated cheaply. The remaining edges in the network are operated at regular cost. The resulting problem is to determine the set of nodes to open hubs and the set of edges to establish in order to find a network of minimum total cost.We consider the case where the network is required to form a Steiner tree spanning a given set of terminal vertices. When edge costs are non-metric, we show logarithmic approximation hardness even for the special case of spanning trees. On the other hand, we show a polynomial-time reduction for Steiner trees to its corresponding node-weighted version thus proving a logarithmic approximation factor. When edge costs are metric, we show the problem is only a constant factor harder to approximate than its original version (with no hub installation) using a similar reduction.