The survivable network design problem (SNDP) is the following problem: given an undirected graph and values Ö for each pair of vertices and , find a minimum-cost subgraph such that there are Ö disjoint paths between vertices and . 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. 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 Ö are only specified for pairs of terminals , and the paths from to must be element disjoint. Thus if Ö ½ elements fail, terminals and 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 ¼ ½ -vertex connectivity, and the 2-approximation algorithm for ¼ ½ ¾ -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 Ç´ÐÓ µ-approximation algorithm, where Ñ Ü Ö . VC-SNDP is not known to have a non-trivial approximation algorithm. In this paper we investigate applying iterative rounding to ELC and VC-SNDP. 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 VC-SNDP when Ö ¾ ¼ ½ ¾ . The latter result improves on an existing 3-approximation algorithm. The former is the first constant factor approximation algorithm for a general survivable network design problem that allows node failures.
There are 2-approximation algorithms for 2-ST and 2-SS by Fleischer et al. REF using the iterative rounding method.
Iterative rounding 2-approximation algorithms for minimum-cost vertex connectivity problems
J. Comput. Syst. Sci.
Mathematics Computer Science