  We present the first nontrivial approximation algorithm for the bottleneck
asymmetric traveling salesman problem. Given an asymmetric metric cost between
n vertices, the problem is to find a Hamiltonian cycle that minimizes its
bottleneck (or maximum-length edge) cost. We achieve an O(log n / log log n)
approximation performance guarantee by giving a novel algorithmic technique to
shortcut Eulerian circuits while bounding the lengths of the shortcuts needed.
This allows us to build on a related result of Asadpour, Goemans, M\k{a}dry,
Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our
technique yields stronger approximation bounds in some cases, such as the
bounded orientable genus case studied by Oveis Gharan and Saberi. We also
explore the possibility of further improvement upon our main result through a
comparison to the symmetric counterpart of the problem.
