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 ε, in polynomial time we construct a sub graph that preserves all pairwise distances between terminals up to a multiplicative 1+ε factor, of total weight at most Oε(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 εD. Namely, given a minor-free graph G = (V, E, w) of diameter D, and parameter ε, we construct a distribution D over dominating metric embeddings into treewidth- Oε(logn) graphs such that ∀u, v ∈ V, \mathbbEf ~ D[dH(f(u), f(v))] ≤ dG(u, v)+εD. 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).