路可分图的度量分解
Metric Decompositions of Path-Separable Graphs
摘要：
在许多涉及度量空间的问题中，一个重要的工具是随机低直径分解的概念。不严格地说，$β$分解是指将度量划分为低直径集合的概率分布，使得附近的点（由$β>0$参数化）可能被“聚类”在一起。将此概念应用于边加权图中的最短路径度量，已知$n$ -顶点图允许$O(\ln n)$ -填充分解（Bartal，1996），并且排除子图允许$O(1)$ -填充分解（Klein，Plotkin and Rao 1993，Fakcharoenphol and Talwar 2003，Abraham et al. 2014）。我们设计了$p$ -路可分图族的分解，这是由Abraham和Gavoille（2006）定义的。并且指的是允许顶点分隔符在图中至多由$p$最短路径组成的图。我们的主要结果是：每个$p$ -路可分的$n$ -顶点图都有一个$O(\ln (p \ln n))$ -分解，它改进了一般图的$O(\ln n)$界，并为有界树宽图等图族提供了新的界.从技术上讲，我们的聚类过程与以前的不同之处在于，我们在精心选择的子图（的最短路径度量）中工作。
英文摘要：
A prominent tool in many problems involving metric spaces is a notion of randomized low-diameter decomposition. Loosely speaking, $β$-decomposition refers to a probability distribution over partitions of the metric into sets of low diameter, such that nearby points (parameterized by $β>0$) are likely to be "clustered" together. Applying this notion to the shortest-path metric in edge-weighted graphs, it is known that $n$-vertex graphs admit an $O(\ln n)$-padded decomposition (Bartal, 1996), and that excluded-minor graphs admit $O(1)$-padded decomposition (Klein, Plotkin and Rao 1993, Fakcharoenphol and Talwar 2003, Abraham et al. 2014). We design decompositions to the family of $p$-path-separable graphs, which was defined by Abraham and Gavoille (2006). and refers to graphs that admit vertex-separators consisting of at most $p$ shortest paths in the graph. Our main result is that every $p$-path-separable $n$-vertex graph admits an $O(\ln (p \ln n))$-decomposition, which refines the $O(\ln n)$ bound for general graphs, and provides new bounds for families like bounded-treewidth graphs. Technically, our clustering process differs from previous ones by working in (the shortest-path metric of) carefully chosen subgraphs.
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