  We present a new $(\frac32+\frac1{\mathrm{e}})$-approximation algorithm for
the Ordered Traveling Salesperson Problem (Ordered TSP). Ordered TSP is a
variant of the classical metric Traveling Salesperson Problem (TSP) where a
specified subset of vertices needs to appear on the output Hamiltonian cycle in
a given order, and the task is to compute a cheapest such cycle. Our
approximation guarantee of approximately $1.868$ holds with respect to the
value of a natural new linear programming (LP) relaxation for Ordered TSP. Our
result significantly improves upon the previously best known guarantee of
$\frac52$ for this problem and thereby considerably reduces the gap between
approximability of Ordered TSP and metric TSP. Our algorithm is based on a
decomposition of the LP solution into weighted trees that serve as building
blocks in our tour construction.
