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Issue No. 07 - July (2012 vol. 61)
ISSN: 0018-9340
pp: 954-968
Hong Shen , University of Adelaide, Adelaide
Longkun Guo , Univ. of Science & Technology of China, Hefei
For an undirected and weighted graph G=(V,E) and a terminal set S\subseteq 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 , 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(\vert V\vert^{2}\vert S\vert^{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(\vert V\vert^{5}\vert E\vert^{7}) for strongly polynomial time and O(\vert V\vert^{5}\gamma ) for weakly polynomial time where \gamma 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(\vert V\vert^{2}\vert S\vert^{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.
Survivable network design, 2-vertex (edge) connected Steiner minimal network, terminal spanning-tree, approximation algorithm, shortest disjoint path pair, Euler walk.

L. Guo and H. Shen, "Efficient 2-Approximation Algorithms for Computing 2-Connected Steiner Minimal Networks," in IEEE Transactions on Computers, vol. 61, no. , pp. 954-968, 2011.
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