  Understanding the structure of minor-free metrics, namely shortest path
metrics obtained over a weighted graph excluding a fixed minor, has been an
important research direction since the fundamental work of Robertson and
Seymour. A fundamental idea that helps both to understand the structural
properties of these metrics and lead to strong algorithmic results is to
construct a "small-complexity" graph that approximately preserves distances
between pairs of points of the metric. We show the two following structural
results for minor-free metrics:
  1. Construction of a light subset spanner. Given a subset of vertices called
terminals, and $\epsilon$, in polynomial time we construct a subgraph that
preserves all pairwise distances between terminals up to a multiplicative
$1+\epsilon$ factor, of total weight at most $O_{\epsilon}(1)$ times the weight
of the minimal Steiner tree spanning the terminals.
  2. Construction of a stochastic metric embedding into low treewidth graphs
with expected additive distortion $\epsilon D$. Namely, given a minor free
graph $G=(V,E,w)$ of diameter $D$, and parameter $\epsilon$, we construct a
distribution $\mathcal{D}$ over dominating metric embeddings into
treewidth-$O_{\epsilon}(\log n)$ graphs such that the additive distortion is at
most $\epsilon D$.
  One of our important technical contributions is a novel framework that allows
us to reduce \emph{both problems} to problems on simpler graphs of bounded
diameter. Our results have the following algorithmic consequences: (1) the
first efficient approximation scheme for subset TSP in minor-free metrics; (2)
the first approximation scheme for vehicle routing with bounded capacity in
minor-free metrics; (3) the first efficient approximation scheme for vehicle
routing with bounded capacity on bounded genus metrics.
