A 2.5-factor approximation algorithm for the k-MST problem

The k-MST problem requires finding that subset of at least k vertices of a given graph whose Minimum Spanning Tree has least weight amongst all subsets of at least k vertices. There has been much work on this problem recently, culminating in an approximation algorithm by Garg, which finds a subset of k vertices whose MST has weight at most 3 times the optimal. Garg also argued that a factor of 3 cannot be improved unless lower bounds different from his are used. This argument applies only to the rooted case of the problem. When no root vertex is specified, we show how to use a pruning technique on top of Garg's algorithm to achieve an approximation factor of 2.5. Note that Garg's algorithm is based upon the Goemans-Williamson clustering method, using which it seems hard to obtain any approximation factor better than 2.