Approximation concerning Asymmetric TSP, Symmetric TSP, and Metric TSP
Here is an excerpt from the introduction to an earlier paper of Svensson, Tarnawski and Végh, which was the first to give a constant factor approximation algorithm for ATSP:

Without any assumptions on the distances, a simple reduction from the problem
of deciding whether a graph is Hamiltonian shows that it is NP-hard to approximate
the shortest tour to within any factor. Therefore it is common to relax the problem by
allowing the tour to visit cities more than once. This is equivalent to assuming that the
distances satisfy the triangle inequality: the distance from city $i$ to $k$ is no larger than
the distance from $i$ to $j$ plus the distance from $j$ to $k$. All results mentioned and proved
in this paper refer to this setting.