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, Halld\'orsson and Hassin \cite{AHH} give approximation algorithms with factors respectively 3.5 and 5.5. Later K\"onemann, Konjevod, Parekh, and Sinha \cite{KKPS} 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 $D \subseteq E$ of $G$. %For the tree and tour covers problems, %In the first paper, in their conclusion %the authors remark that their algorithms do not work for the generalized tree and tours covers problems and %these problems have not been considered in the second paper. We show that the algorithms of K\"onemann et al. can be adapted for the generalized tree and tours covers problem with the same factors.

## Citation

accepted for publication in RAIRO

## Article

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