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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关于 Light Spanner、低树宽嵌入和 Minor-free 图中的高效遍历

自 Robertson 和 Seymour 的基础工作以来，了解无次要度量的结构，即在不包括固定次要的加权图上获得的最短路径度量，一直是一个重要的研究方向。一个有助于理解这些度量的结构特性并产生强大算法结果的基本思想是构建一个“小复杂性”图，该图近似保留度量的点对之间的距离。我们展示了无次要指标的以下两个结构结果： 1. 轻量子集扳手的构建。给定一个称为终端的顶点子集 $\epsilon$，在多项式时间内，我们构造一个子图，该子图保留终端之间的所有成对距离，直到乘法 $1+\epsilon$ 因子，总权重至多为 $O_{\epsilon}(1)$ 乘以跨越终端的最小 Steiner 树的权重。2. 构建嵌入到具有预期加性失真 $\epsilon D$ 的低树宽图中的随机度量。也就是说，给定一个直径为 $D$ 的次要自由图 $G=(V,E,w)$ 和参数 $\epsilon$，我们构建了一个分布 $\mathcal{D}$ 在支配度量嵌入到 treewidth-$ O_{\epsilon}(\log n)$ 图形使得加性失真至多为 $\epsilon D$。我们的一项重要技术贡献是一个新颖的框架，它使我们能够将 \emph{both questions} 减少到更简单的有界直径图上的问题。我们的结果具有以下算法结果：（1）子集 TSP 在次要自由度量中的第一个有效近似方案；(2) 在minor-free度量中具有有界容量的车辆路由的第一个近似方案；(3) 第一个在有界属度量上具有有界容量的车辆路由的有效近似方案。