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Proceedings 2001 IEEE International Conference on Cluster Computing (2001)
Las Vegas, Nevada
Oct. 14, 2001 to Oct. 17, 2001
ISBN: 0-7695-1390-5
pp: 339
<p>In the survivable network design problem (SNDP), given an undirected graph and values r<sub>ij</sub> for each pair of vertices i and j, we attempt to find a minimum-cost subgraph such that there are r<sub>ij</sub> disjoint paths between vertices i and j. In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. Jain et al. [12] propose a version of the problem intermediate in difficulty to these two, called the element connectivity problem (ELC-SNDP, or ELC). In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values r<sub>ij</sub> are only specified for pairs of terminals i, j, and the paths from i to j must be element disjoint. Thus if r<sub>ij</sub> − 1 elements fail, terminals i and j are still connected by a path in the network.</p> <p>These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 (due to Jain [11]), and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O(log k)-approximation algorithm, where k = max<sub>i,j</sub> r<sub>ij</sub> (Jain et al. [12]). VC-SNDP is not known to have a non-trivial approximation algorithm; however, recently Fleischer [7] has shown how to extend the technique of Jain [11] to give a 2-approximation algorithm in the case that r <sub>ij</sub> \in? {0, 1, 2}. She also shows that the same techniques will not work for VC-SNDP for more general values of r<sub>ij</sub>.</p> <p>In this paper we show that these techniques can be extended to a 2-approximation algorithm for ELC. This gives the first constant approximation algorithm for a general survivable network design problem which allows node failures.</p>

L. Fleischer, D. Williamson and K. Jain, "An Iterative Rounding 2-Approximation Algorithm for the Element Connectivity Problem," Proceedings 2001 IEEE International Conference on Cluster Computing(FOCS), Las Vegas, Nevada, 2001, pp. 339.
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