Approximate k-MSTs and k-steiner trees via the primal-dual method and Lagrangean relaxation

Abstract

We consider the problem of computing the minimum-cost tree spanning at least k vertices in an undirected graph. Garg [10] 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 [15], 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.