Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems

The survivable network design problem (SNDP) is the following problem: given an undirected graph and values rij for each pair of vertices i and j, find a minimum-cost subgraph such that there are rij disjoint paths between vertices i and j. In the edge connected version of this problem (ECSNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. The element connectivity problem (ELC-SNDP, or ELC) is a problem of intermediate difficulty. 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 rij are only specified for pairs of terminals i; j, and the paths from i to j must be element disjoint. Thus if rij 1 elements fail, terminals i and j are still connected by a path in the network. 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 This paper is the union of two previously published extended abstracts. The extended abstract [5] presents negative examples for f0; 1; : : : ; kg-vertex connectivity, and the 2-approximation algorithm for f0; 1; 2g-vertex connectivity. This result is generalized to include element connectivity in [6]. and iteratively rounds solutions to a linear programming relaxation of the problem. ELC has a primal-dual O(log k)-approximation algorithm, where k = maxi;j rij . VC-SNDP is not known to have a non-trivial approximation algorithm. In this paper we investigate applying iterative rounding to ELC and VCSNDP. We show that iterative rounding will not yield a constant factor approximation algorithm for general VC-SNDP. On the other hand, we show how to extend the analysis of iterative rounding applied to EC-SNDP to yield 2-approximation algorithms for both general ELC, and for the case of VCSNDP when rij 2 f0; 1; 2g. The latter result improves on an existing 3approximation algorithm. The former is the first constant factor approximation algorithm for a general survivable network design problem that allows node failures.