## 1 Introduction  
Let \(G=(V,A)\) be a directed graph with a (non-negative) cost function \(c:A\Rightarrow\mathbb{Q}_{+}\) defined on the arcs. A _directed tour cover_\(T\) is a subgraph \(T=(U,F)\) such that  
1. For every \(e\in A\), \(F\) contains an arc \(f\) intersecting \(e\), _i.e._, \(f\) and \(e\) have at least one end-node in common. 2. \(T\) is a closed directed walk (or branching in the case of tree cover). We consider in this paper _the minimum directed tour cover (DToCP)_ problem which is to find a directed tour cover of minimum cost. As in the previous works [1,2] on the undirected tour cover, we allow the directed tour cover to be a closed directed walk, _i.e._, it may visit a node more than once, to avoid restricting the cost function \(c\) to be a metric. However, if the cost function \(c\) satisfies the triangle inequality, this directed walk can be short-cut into a simple directed cycle which covers every arc in \(A\) without increasing the cost. The undirected version of DToCP is the minimum tour cover (ToCP) problem which has been studied in recent years. The ToCP was introduced in a paper by [PERSON] _et al._[1]. The authors proved that ToCP is \(NP\)-hard and gave a purely combinatorial 5.5-approximation algorithm. Later, [PERSON] _et al._[2] proposed an integer formulation for ToCP and used it to improve the approximation ratio to 3. In this paper, the DToCP was listed as an open problem. We can prove that DToCP is \(NP\)-hard by a reduction from the metric Asymmetric Traveling Salesman Problem (ATSP). The proof is very similar to the one for ToCP described in [1]. Given any instance \(G=(V,A)\) of ATSP, for every node \(v\in V\), we create a new node \(v^{\prime}\) and an arc \((v,v^{\prime})\) of cost \(+\infty\) to obtain a new graph \(G^{\prime}\). Let us consider the DToCP on \(G^{\prime}\), we can see that any directed tour cover on \(G^{\prime}\) must cover \(V\). As the costs are metric, given any directed tour cover, we can find a traveling salesman tour on \(G\) of no greater cost. Hence, there is an one-to-one correspondence between the optimal solutions of DToCP on \(G^{\prime}\) and the optimal traveling salesman tours on \(G\). Moreover, this reduction is approximation-preserving, _i.e._, if we have an \(\alpha\)-approximation algorithm for DToCP then we can approximate the metric ATSP with the same ratio \(\alpha\). In our knowledge, the best approximation ratio for ATSP is so far \(0.824\log_{2}(n)\) achieved by [PERSON] _et al._[3]. Therefore, it is rather difficult to design an algorithm of a constant approximation ratio for DToCP. In this paper, we will present a \(2\log_{2}(n)\)-approximation algorithm for the DToCP. This ratio is more than 2 times the ratio for ATSP, since our algorithm is inspired from the algorithm given by [PERSON] _et al._[2] for the undirected case which is based on the bound of the Held-Karp relaxation for metric TSP. It is proved that the metric TSP costs no more than \(\frac{3}{2}\) the Held-Karp bound for the undirected case and the metric ATSP costs no more than \(\log_{2}(n)\) the Held-Karp bound for the directed case [4, 5]. Hence, our ratio \(2\log_{2}(n)\) for DToCP follows logically the best ratio \(3\) for ToCP. Our work for DToCP is motivated by the asymmetric versions of the applications enumerated by [PERSON] _et al._ for the undirected case, especially a special case of the Symmetric Traveling Purchaser Problem [1]. We give here a concrete example of applications of DToCP, which is the following simplified version of the waste collection problem. A company is in charge of waste collection in an urban area. The area is supposed to be small enough so that one truck is sufficient to collect all of its daily wastes. The gather is done by the truck in early morning of each day. For each street in the area, waste should be regrouped at one of the end-points of the street in the night of the day before. When the truck goes along a street, this induces some cost to the company.