An Approximation Algorithm for the Two-Node-Connected Star Problem with Steiner Nodes

The goal in topological network design is to build a minimum-cost topology meeting specific real-life constraints. There is a cost-robustness trade-off under single and multiple failures.

Previous works in the field suggest that a backbone composed by a two-node-connected toplogy provides savings with respect to elementary cycles. Consequently, we introduce the Two-Node Connected Star Problem with Steiner Nodes (2NCSP-SN). The goal is to design a minimum-cost topology, where the backbone is two-node connected, the access network is connected in a star topology or by direct links to the backbone, and optional nodes (called Steiner nodes) could be included in the solution. The 2NCSP-SN belongs to the class of NP-Hard problems. This promotes the development of heuristics and approximation algorithms.

An approximation algorithm of factor 4α for the 2NCSP-SN is introduced, being $\alpha \ge 1/2$ the cost-ratio between backbone and access links. This is a generalization of the well-known factor 2 for the design of minimum-cost two-connected spanning networks (if we fix $\alpha =1/2$). Finally, an exact Integer Linear Programming (ILP) formulation is proposed in order to highlight the effectiveness of the approximation algorithm. The results confirm a small gap between the globally optimum solution and the topology offered by our approximation algorithm when the ratio α is close to 1/2.