  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 $\frac{7}{10}$-approximation algorithm for Max ATSP. It is
based on techniques of {\em eliminating} and {\em diluting} problematic
subgraphs with the aid of {\it half-edges} and a method of edge coloring. (A
{\it half-edge} of edge $(u,v)$ is informally speaking "either a head or a tail
of $(u,v)$".) A novel technique of {\em diluting} a problematic subgraph $S$
consists in a seeming reduction of its weight, which allows its better
handling.
  The current best approximation algorithms for Max ATSP, achieving the
approximation guarantee of $\frac 23$, are due to Kaplan, Lewenstein, Shafrir,
Sviridenko (2003) and Elbassioni, Paluch, van Zuylen (2012). Using a result by
Mucha, which states that an $\alpha$-approximation algorithm for Max ATSP
implies a $(2+\frac{11(1-\alpha)}{9-2\alpha})$-approximation algorithm for the
shortest superstring problem (SSP), we obtain also a $(2 \frac{33}{76} \approx
2,434)$-approximation algorithm for SSP, beating the previously best known
(having an approximation factor equal to $2 \frac{11}{23} \approx 2,4782$.)
