Approximation algorithms for metric tree cover and generalized tour and tree covers

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, Halldórsson and Hassin (1993) give approximation algorithms with factors respectively 3.5 and 5.5. Later Könemann, Konjevod, Parekh, and Sinha (2003) study the linear programming relaxations and improve 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.