Efficient 2-Approximation Algorithms for Computing 2-Connected Steiner Minimal Networks

For an undirected and weighted graph G = (V, E) and a terminal set S ⊆ V , the 2-connected Steiner minimal network (SMN) problem requires to compute a minimum-weight subgraph of G in which all terminals are 2-connected to each other. This problem has important applications in design of survivable networks and fault-tolerant communication, and is known MAXSNP-hard [7], a harder subclass of NP-hard problems for which no polynomial-time approximation scheme (PTAS) is known. This paper presents an efficient algorithm of O(|V|²|S|³) time for computing a 2-vertex connected Steiner network (2VSN) whose weight is bounded by two times of the optimal solution 2-vertex connected SMN (2VSMN). It compares favorably with the currently known 2-approximation solution to the 2VSMN problem based on that to the survivable network design problem [10], [16], with a time complexity reduction of O(|V|⁵|E|⁷) for strongly polynomial time and O(|V|⁵γ) for weakly polynomial time where -y is determined by the sizes of input. Our algorithm applies a novel greedy approach to generate a 2VSN through progressive improvement on a set of vertex-disjoint shortest path pairs incident with each terminal of S. The algorithm can be directly deployed to solve the 2-edge connected SMN problem at the same approximation ratio within time O(|V|²|S|²). To the best of our knowledge, this result presents currently the most efficient 2-approximation algorithm for the 2-connected Steiner minimal network problem.