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Given a directed graph _{2}(

Let _{+} defined on the arcs. A

For every

We consider in this paper

We can prove that DToCP is _{2}(_{2}(_{2}(_{2}(

The paper is organized as follows: In Section 2, we present an integer formulation for DToCP. We discuss in Section 3 about the parsimonious property of Eulerian directed graphs, in particular in Section 3.1. We prove the equivalent directed version of Goemans and Bertsimas's theorem on splitting operations. In Section 3.2, we derive an equivalent linear program to the Held-Karp relaxation for metric ATSP. This linear program will be useful in the analysis of our algorithm's performance guarantee in Section 5. Before that, we state our approximation algorithm for DToCP in Section 4.

Let us introduce the notations that will be used in the paper. Let ^{|}^{A}^{|} is a vector indexed by the arc set _{e}_{∈}_{F} x_{e}^{|}^{V}^{|} indexed by the vertices and ^{+}(^{−}(^{+}(^{−}(^{+}(^{−}(^{+}(^{−}^{+}(

Let

Given any tour cover ^{+}(^{−}(^{−}(^{+}(_{e}

If one specific vertex _{e}_{∈}_{A} c_{e}x_{e}_{e}_{1}, _{2} ∈ ^{+}(

Given a positive integer

^{+}(

^{−}(

^{+}(

_{e}

Let us consider the linear programming relaxation of (

^{+}(

^{−}(

^{+}(

_{e}

The parsimonious property for Eulerian directed graphs says that if the costs are metric, in (

^{+}(

^{+}(

^{−}(

^{+}(

_{e}

is equivalent to (

If the costs

The parsimonious property can be also formulated for undirected Eulerian graphs. We simply replace the outgoing and ingoing cuts in (

The undirected version of Theorem 1 (in a little more general form) has been proved by Goemans and Bertsimas [

[_{G}

_{G′}_{G}

_{G′}_{G}_{G}_{G}

Condition 1 of Lemma 1 is a result due to Lovasz [

[

_{G′}_{G}

Like the work of Goemans and Bertsimas for the undirected case, we will subsequently prove that we can add a similar condition as Condition 2 of Lemma 1 to Lemma 2 which express the connectivities involving

Let

_{G′}_{G}

Our proof for Lemma 3 is very similar to the Goemans and Bertsimas' proof for Lemma 1. This latter is also in major part inspired from the one of Lovasz for Lemma 1 (Condition 1) which proceeds along the following lines.

There exists at most one set

_{g}_{G}

If there is no such

If such a

We will show that this proof procedure can be used for the directed case. Let us remove the orientation on the arcs of _{G}_{G}_{G}

Let _{G}

_{G}_{G}_{G}

We have
_{G}^{+}(_{G}^{−}(_{G}_{G}_{G}_{G}_{G}

Hence, given any

either there exists exactly a

Indeed, by Remark 1, if

The conditions

otherwise, _{G}_{G}_{G}

Hence we can summarize below the directed version of Lovasz's proof procedure.

There exists at most one set

If there is no such

If such a

To show Lemma 3, we add a new node

Now applying the proof procedure to

There is no _{G′}_{Ĝ}

As _{Ĝ}_{G′}_{G′}_{G′}_{g}_{G′}

Otherwise,

_{Ĝ}

Hence, there would exist some _{Ĝ}_{Ĝ}

Therefore, there exists some _{Ĝ}_{Ĝ′}

Moreover, due to splitting operation _{G′}_{G}

We finish here the proof of Lemma 3.

Let _{e∈A} _{e}x_{e}_{e}_{e}^{+}(

_{G′}

Let

Let us consider the Asymmetric Traveling Salesman Problem (ATSP) on

^{+}(

^{+}(

^{−}(

_{e}

This linear program is called _{e}^{+}(^{−}(

^{+}(

^{+}(

^{−}(

_{e}

The following theorem is a direct application of Theorem 1.

If the costs

^{+}(

^{+}(

^{−}(

_{e}

We are now ready to state our algorithm for directed tour cover which is heavily inspired from the one for ToCP in [

Let

Let

Let _{U}_{U}, A_{U}_{U}_{U}_{U}

Run the Frieze, Galbiati and Maffioli heuristic [_{U}^{+}({^{+}(^{+}(

Let x* be the vector minimizing Σ_{e∈A} _{e}x_{e}_{G̃}

_{G̃}

To prove the remark, we will show that for any

In _{U}_{U}

After these splitting operations, _{G̃U}

Consider the graph _{U}_{U}_{U}_{U}_{U}_{U}^{U}^{|AU|} such that for each _{U}_{e}_{U}_{e}_{U}

We have 2^{U}x^{U}

We have seen that ^{U}^{U}^{U}^{U}^{U}x^{U}

Let _{U}

^{U}x^{U}^{U}z

We can see that ^{U}_{U}^{U} x^{U}x^{U}^{U}z

We are ready now to state the main theorem

The algorithm outputs a directed tour cover of cost no more than 2 log_{2}(

The algorithm return, as a solution of DToCP, the traveling salesman directed tour output _{U}^{T}^{U}x^{T}_{2}(_{U}, i.e., c^{U}x^{T}_{2}(^{U}z^{U}x^{T}_{2}(

In a recent paper [^{20}) than log_{2}(