  We prove the approximation ratio 8/5 for the metric $\{s,t\}$-path-TSP
problem, and more generally for shortest connected $T$-joins.
  The algorithm that achieves this ratio is the simple "Best of Many" version
of Christofides' algorithm (1976), suggested by An, Kleinberg and Shmoys
(2012), which consists in determining the best Christofides $\{s,t\}$-tour out
of those constructed from a family $\Fscr_{>0}$ of trees having a convex
combination dominated by an optimal solution $x^*$ of the fractional
relaxation. They give the approximation guarantee $\frac{\sqrt{5}+1}{2}$ for
such an $\{s,t\}$-tour, which is the first improvement after the 5/3 guarantee
of Hoogeveen's Christofides type algorithm (1991). Cheriyan, Friggstad and Gao
(2012) extended this result to a 13/8-approximation of shortest connected
$T$-joins, for $|T|\ge 4$.
  The ratio 8/5 is proved by simplifying and improving the approach of An,
Kleinberg and Shmoys that consists in completing $x^*/2$ in order to dominate
the cost of "parity correction" for spanning trees. We partition the edge-set
of each spanning tree in $\Fscr_{>0}$ into an $\{s,t\}$-path (or more
generally, into a $T$-join) and its complement, which induces a decomposition
of $x^*$. This decomposition can be refined and then efficiently used to
complete $x^*/2$ without using linear programming or particular properties of
$T$, but by adding to each cut deficient for $x^*/2$ an individually tailored
explicitly given vector, inherent in $x^*$.
  A simple example shows that the Best of Many Christofides algorithm may not
find a shorter $\{s,t\}$-tour than 3/2 times the incidentally common optima of
the problem and of its fractional relaxation.
