In the maximum asymmetric traveling salesman problem (Max ATSP) we are given a complete directed graph with nonnegative weights on the edges and we wish to compute a traveling salesman tour of maximum weight. In this paper we give a fast combinatorial $\frac 34$-approximation algorithm for Max ATSP. It is based on a novel use of {\it half-edges}, matchings and a new method of edge coloring. (A {\it half-edge} of edge $(u,v)$ is informally speaking "either a head or a tail of $(u,v)$".) The current best approximation algorithms for Max ATSP, achieving the approximation guarantee of $\frac 23$, are due to Kaplan, Lewenstein, Shafrir and Sviridenko and Elbassioni, Paluch, van Zuylen. Using a recent result by Mucha, which states that an $α$-approximation algorithm for Max ATSP implies a $(2+\frac{11(1-α)}{9-2α})$-approximation algorithm for the shortest superstring problem (SSP), we obtain also a $(2 \frac{11}{30} \approx 2,3667)$-approximation algorithm for SSP, beating the previously best known (having approximation factor equal to $2 \frac{11}{23} \approx 2,4782$.)