## Abstract.

Garg [10] gives two approximation algorithms for the minimum-cost tree spanning *k* vertices in an undirected graph. Recently Jain and Vazirani [15] discovered primal-dual approximation algorithms for the metric uncapacitated facility location and *k*-median problems. In this paper we show how Garg’s algorithms can be explained simply with ideas introduced by Jain and Vazirani, in particular via a Lagrangean relaxation technique together 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.

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Chudak, F., Roughgarden, T. & Williamson, D. Approximate *k*-MSTs and *k*-Steiner trees via the primal-dual method and Lagrangean relaxation.
*Math. Program., Ser. A* **100**, 411–421 (2004). https://doi.org/10.1007/s10107-003-0479-2

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DOI: https://doi.org/10.1007/s10107-003-0479-2