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How to trim a MST: A 2-Approximation algorithm for minimum cost-tree cover

The minimum cost-tree cover problem is to compute a minimum cost-tree T in a given connected graph G with costs on the edges, such that the vertices spanned by T form a vertex cover for G. The problem is supposed to occur in applications of vertex cover and in edge-dominating sets when additional connectivity is required for solutions. Whereas a linear-time 2-approximation algorithm for the unweighted case has been known for quite a while, the best approximation ratio known for the weighted case is 3. Moreover, the 3-approximation algorithms for such cases are far from practical due to their inefficiency.
 In this article we present a fast, purely combinatorial 2-approximation algorithm for the minimum cost-tree cover problem. It constructs a good approximate solution by trimming some leaves within a minimum spanning tree (MST); and, to determine which leaves to trim, it uses both the primal-dual schema and an instance layering technique adapted from the local ratio method.