On light spanners, low-treewidth embeddings and efficient traversing in minor-free graphs

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 sub graph 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 forall u, v in V, mathbb{E} {f sim mathcal{D}}[d {H}(f(u), f(v))] leq d {G}(u, v)+ epsilon D. 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 bounded-capacity vehicle routing in minor-free metrics; (3) the first efficient approximation scheme for bounded-capacity vehicle routing on bounded genus metrics. En route to the latter result, we design the first FPT approximation scheme for bounded-capacity vehicle routing on bounded-treewidth graphs (parameterized by the treewidth).