Reduction from TSP to ATSP does not imply constant factor approximation algorithm
I think the reduction they showed is incorrect. Where it falls is when you connect a node to its &quot;ghost&quot; with 0 cost and add a positive $w$ to all other edges. We will take a look why such a behavior can change the optimal path.
Indeed, take a look at a graph with nodes $v_0,\dots,v_{10}$ where we want to find the shortest path between $v_0$ and $v_{10}$.
The graph's edges will connect $v_i$ to $v_{i+1}$ with weight $1$, and will additionally connect $v_0$ to $v_{10}$ with weight $15$.
Clearly, the path between $v_0$ to $v_1$ and so on until $v_{10}$ has the lowest cost, of $10$.
But now assume we added to all edges a weight $w=1$. Then, this path's value will double to $20$, and the edge between $v_1$ and $v_{10}$ will have weight $16$.
Obviously, the shortest path now is the direct path $v_0,v_{10}$ with cost $16$, hence the optimal path changed.
Important note: Its possible to find a small enough $w$ that will not change the optimal path, but I believe finding one is NP-hard.