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

Abstract

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 OVj2jSj3Þ time for computing a 2-vertex connected Steiner network (2V SNÞ whose weight is bounded by two times of the optimal solution 2-vertex connected SMN (2V SMNÞ. It compares favorably with the currently known 2-approximation solution to the 2V SMN problem based on that to the survivable network design problem [10], [16], with a time complexity reduction of OðjV j5jEj7Þ for strongly polynomial time and OðjV j5_Þ for weakly polynomial time where _ is determined by the sizes of input. Our algorithm applies a novel greedy approach to generate a 2V SN 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 OðjV j2jSj2Þ. To the best of our knowledge, this result presents currently the most efficient 2-approximation algorithm for the 2-connected Steiner minimal network problem.