# Erratum: An Approximation Algorithm for Minimum-Cost Vertex-Connectivity Problems

## Abstract

There is an error in our paper ``An Approximation Algorithm for Minimum-Cost Vertex- Connectivity Problems'' (*Algorithmica* (1997), 18:21—43). In that paper we considered the following problem: given an undirected graph and values *r*_{ ij } for each pair of vertices *i* and *j* , find a minimum-cost set of edges such that there are *r*_{ ij } vertex-disjoint paths between vertices *i* and *j* . We gave approximation algorithms for two special cases of this problem. Our algorithms rely on a primal—dual approach which has led to approximation algorithms for many edge-connectivity problems. The algorithms work in a series of stages; in each stage an augmentation subroutine augments the connectivity of the current solution. The error is in a lemma for the proof of the performance guarantee of the augmentation subroutine.

In the case *r*_{ ij }* = k* for all *i,j* , we described a polynomial-time algorithm that claimed to output a solution of cost no more than 2 **H** *(k)* times optimal, where **H** = 1 + 1/2 + · · · + 1/*n* . This result is erroneous. We describe an example where our primal—dual augmentation subroutine, when augmenting a *k* -vertex connected graph to a *(k+1)* -vertex connected graph, gives solutions that are a factor *Ω(k)* away from the minimum.

In the case *r*_{ ij }* ∈ {0,1,2}* for all *i,j* , we gave a polynomial-time algorithm which outputs a solution of cost no more than three times the optimal. In this case we prove that the statement in the lemma that was erroneous for the *k* -vertex connected case does hold, and that the algorithm performs as claimed.

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