  Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic
embedding of minor-free graphs of diameter $D$ into graphs of treewidth
$O_{\epsilon}(\log n)$ with expected additive distortion $+\epsilon D$.
Cohen-Addad et al. then used the embedding to design the first quasi-polynomial
time approximation scheme (QPTAS) for the capacitated vehicle routing problem.
Filtser and Le [STOC'21] used the embedding (in a different way) to design a
QPTAS for the metric Baker's problems in minor-free graphs. In this work, we
devise a new embedding technique to improve the treewidth bound of Cohen-Addad
et al. exponentially to $O_{\epsilon}(\log\log n)^2$. As a corollary, we obtain
the first efficient PTAS for the capacitated vehicle routing problem in
minor-free graphs. We also significantly improve the running time of the QPTAS
for the metric Baker's problems in minor-free graphs from
$n^{O_{\epsilon}(\log(n))}$ to $n^{O_{\epsilon}(\log\log(n))^3}$.
  Applying our embedding technique to planar graphs, we obtain a deterministic
embedding of planar graphs of diameter $D$ into graphs of treewidth
$O((\log\log n)^2)/\epsilon)$ and additive distortion $+\epsilon D$ that can be
constructed in nearly linear time. Important corollaries of our result include
a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle
routing problem with bounded capacity in planar graphs, both run in
almost-linear time. The running time of our algorithms is significantly better
than previous algorithms that require quadratic time.
  A key idea in our embedding is the construction of an (exact) emulator for
tree metrics with treewidth $O(\log\log n)$ and hop-diameter $O(\log \log n)$.
This result may be of independent interest.
