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.
翻译:了解不光量度的结构, 也就是在不固定的平面上获得的最短路径度量, 自罗伯逊和赛摩尔的基本工作以来, 一直是一个重要的研究方向。 一个有助于理解这些计量的结构性属性并导致强烈的算法结果的基本想法是构建一个“ 小型复杂度” 图形, 大约能保持两对点之间的距离。 我们展示了以下两个不光度度度度度度量的结构性结果 : 1. 建造一个光子分光仪。 鉴于一个称为终端的脊柱子子子子, 和 $ 和 $ 的 美元, 我们建一个子谱图, 将所有终端之间的双对齐距离保存到一个多倍的 $ 百分数 。 我们的平面值值值值值值值值值值值值值值值 值值值值 。 我们的平面平面值值值值值值值值值值值, 将一个最小的硬度量度量度量度量度量度量度测量器先嵌嵌入低的低树底图, 。