# An approximation algorithm for minimum-cost vertex-connectivity problems

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## Abstract

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 value*r* _{ ij } for each pair of vertices*i* and*j* is given, and a minimum-cost set of edges such that there are*r* _{ ij } vertex-disjoint paths between vertices*i* and*j* must be found. In the case for which*r* _{ ij }∈{0, 1, 2} for all*i, j*, we can find a solution of cost no more than three times the optimal cost in polynomial time. In the case in which*r* _{ ij }=*k* for all*i, j*, we can find a solution of cost no more than 2*H(k)* times optimal, where\(\mathcal{H}(n) = 1 + \tfrac{1}{2} + \cdot \cdot \cdot + \tfrac{1}{n}\). 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.

## Key Words

Approximation algorithm Vertex connectivity Survivable network design Primal-dual method## Preview

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