Given a weighted undirected graph G = (V,E), a tree (respectively tour) cover of an edge-weighted graph is a set of edges which forms a tree (resp. closed walk) and covers every other edge in the graph. The tree (resp. tour) cover problem is of finding a minimum weight tree (resp. tour) cover of G. Arkin et al. gave approximation algorithms with factors respectively 3.5 and 5.5. Later Könemann et al. studied the linear programming relaxations and improved both factors to 3. We describe in the first part of the paper a 2-approximation algorithm for the metric case of tree cover. In the second part, we will consider a generalized version of tree (resp. tour) covers problem which is to find a minimum tree (resp. tours) which covers a subset D⊆E of G. We show that the algorithms of Könemann et al. can be adapted for the generalized tree and tours covers problem with the same factors.
