  We present a fast combinatorial $3/4$-approximation algorithm for the maximum
asymmetric TSP with weights zero and one. The approximation factor of this
algorithm matches the currently best one given by Bl\"aser in 2004 and based on
linear programming. Our algorithm first computes a maximum size matching and a
maximum weight cycle cover without certain cycles of length two but possibly
with {\em half-edges} - a half-edge of a given edge $e$ is informally speaking
a half of $e$ that contains one of the endpoints of $e$. Then from the computed
matching and cycle cover it extracts a set of paths, whose weight is large
enough to be able to construct a traveling salesman tour with the claimed
guarantee.
