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.
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