Cohen-Addad, Le, Pilipczuk, and Pilipczuk [CLPP23] recently constructed a stochastic embedding with expected $1+\varepsilon$ distortion of $n$-vertex planar graphs (with polynomial aspect ratio) into graphs of treewidth $O(\varepsilon^{-1}\log^{13} n)$. Their embedding is the first to achieve polylogarithmic treewidth. However, there remains a large gap between the treewidth of their embedding and the treewidth lower bound of $\Omega(\log n)$ shown by Carroll and Goel [CG04]. In this work, we substantially narrow the gap by constructing a stochastic embedding with treewidth $O(\varepsilon^{-1}\log^{3} n)$. We obtain our embedding by improving various steps in the CLPP construction. First, we streamline their embedding construction by showing that one can construct a low-treewidth embedding for any graph from (i) a stochastic hierarchy of clusters and (ii) a stochastic balanced cut. We shave off some logarithmic factors in this step by using a single hierarchy of clusters. Next, we construct a stochastic hierarchy of clusters with optimal separating probability and hop bound based on shortcut partition [CCLMST23, CCLMST24]. Finally, we construct a stochastic balanced cut with an improved trade-off between the cut size and the number of cuts. This is done by a new analysis of the contraction sequence introduced by [CLPP23]; our analysis gives an optimal treewidth bound for graphs admitting a contraction sequence.

翻译：Cohen-Addad、Le、Pilipczuk 和 Pilipczuk [CLPP23] 最近构造了一种随机嵌入，将具有多项式纵横比的 $n$ 顶点平面图以期望 $1+\varepsilon$ 的失真度嵌入到树宽为 $O(\varepsilon^{-1}\log^{13} n)$ 的图中。他们的嵌入是首个实现多对数树宽的嵌入。然而，其嵌入的树宽与 Carroll 和 Goel [CG04] 证明的 $\Omega(\log n)$ 树宽下界之间仍存在较大差距。在本工作中，我们通过构造一个树宽为 $O(\varepsilon^{-1}\log^{3} n)$ 的随机嵌入，大幅缩小了这一差距。我们通过改进 CLPP 构造中的多个步骤来获得我们的嵌入。首先，我们简化了他们的嵌入构造，证明了对于任意图，可以从（i）一个随机聚类层次结构和（ii）一个随机平衡割来构造一个低树宽嵌入。在此步骤中，我们通过使用单一的聚类层次结构削减了一些对数因子。接着，我们基于捷径划分 [CCLMST23, CCLMST24] 构造了一个具有最优分离概率和跳数界的随机聚类层次结构。最后，我们构造了一个随机平衡割，其在割的规模与割的数量之间实现了改进的权衡。这是通过对 [CLPP23] 引入的收缩序列进行新的分析完成的；我们的分析为允许收缩序列的图给出了最优的树宽界。