  We present a black-box reduction from the path version of the Traveling
Salesman Problem (Path TSP) to the classical tour version (TSP). More
precisely, we show that given an $\alpha$-approximation algorithm for TSP,
then, for any $\epsilon >0$, there is an $(\alpha+\epsilon)$-approximation
algorithm for the more general Path TSP. This reduction implies that the
approximability of Path TSP is the same as for TSP, up to an arbitrarily small
error. This avoids future discrepancies between the best known approximation
factors achievable for these two problems, as they have existed until very
recently.
  A well-studied special case of TSP, Graph TSP, asks for tours in unit-weight
graphs. Our reduction shows that any $\alpha$-approximation algorithm for Graph
TSP implies an $(\alpha+\epsilon)$-approximation algorithm for its path
version. By applying our reduction to the $1.4$-approximation algorithm for
Graph TSP by Seb\H{o} and Vygen, we obtain a polynomial-time
$(1.4+\epsilon)$-approximation algorithm for Graph Path TSP, improving on a
recent $1.497$-approximation algorithm of Traub and Vygen.
  We obtain our results through a variety of new techniques, including a novel
way to set up a recursive dynamic program to guess significant parts of an
optimal solution. At the core of our dynamic program we deal with instances of
a new generalization of (Path) TSP which combines parity constraints with
certain connectivity requirements. This problem, which we call $\Phi$-TSP, has
a constant-factor approximation algorithm and can be reduced to TSP in certain
cases when the dynamic program would not make sufficient progress.
