Efficient 2-approximation algorithms for computing 2-connected steiner minimal networks

For an undirected and weighted graph G=(V,E) and a terminal set SV, 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 , 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 ²S 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], with a time complexity reduction of O(V ⁵E ⁷) for strongly polynomial time and O(V ⁵γ ) for weakly polynomial time where γ 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 ²S 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.