An approximation algorithm for minimum-cost vertex-connectivity problems

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, a valuerij for each pair of verticesi andj is given, and a minimum-cost set of edges such that there arerij vertex-disjoint paths between verticesi andj must be found. In the case for whichrij∈{0, 1, 2} for alli, j, we can find a solution of cost no more than three times the optimal cost in polynomial time. In the case in whichrij=k for alli, j, we can find a solution of cost no more than 2H(k) times optimal, where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{H}(n) = 1 + \tfrac{1}{2} + \cdot \cdot \cdot + \tfrac{1}{n}$$ \end{document}. No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems.