Approximate k-MSTs and k-Steiner Trees via the Primal-Dual Method and Lagrangean Relaxation
We consider the problem of computing the minimum-cost tree spanning at least k vertices in an undirected graph. Garg  gave two approximation algorithms for this problem. We show that Garg’s al- gorithms can be explained simply with ideas introduced by Jain and Vazirani for the metric uncapacitated facility location and k-median problems , in particular via a Lagrangean relaxation technique to- gether with the primal-dual method for approximation algorithms. We also derive a constant-factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.
KeywordsApproximation Algorithm Lagrangean Relaxation Dual Solution Performance Guarantee Root Vertex
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