发表机构
Dartmouth College; University of Massachusetts, Amherst（达特茅斯学院; 马萨诸塞大学阿默斯特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出新洋葱剥皮法，将k-外平面图每次剥去k/2层，首次得到其到树与ℓ₁空间的poly(k)失真嵌入，较此前指数级失真界有指数改进，更接近k-外平面图的失真下界Ω(k)。
AI 中文摘要
Chekuri、Gupta、Newman、Rabinovich与Sinclair在[SODA'03]中证明，k-外平面图可嵌入树与ℓ₁空间，失真为2^{O(k)}。该结果是对仍未解决的平面ℓ₁嵌入猜想（平面度量可嵌入ℓ₁空间，失真为常数）的最强支撑之一，但失真界对k的指数依赖仍是当前最优结果。其嵌入采用所谓洋葱剥皮法：每次移除图的一层，乘法失真代价为O(1)，递归嵌入得到的(k-1)-外平面图。本文提出新的洋葱剥皮法，每次从输入k-外平面图剥去k/2层，首次得到k-外平面图到树与ℓ₁空间的失真为poly(k)的嵌入，较[SODA'03]的最优失真界有指数级改进；该树嵌入结果更接近k-外平面图的失真下界Ω(k)。
英文摘要
Chekuri, Gupta, Newman, Rabinovich, and Sinclair [SODA'03] showed that $k$-outerplanar graphs can be embedded into trees and $\ell_1$ with distortion $2^{O(k)}$. Their result is perhaps the strongest evidence supporting the still-open planar $\ell_1$-embedding conjecture: planar metrics can be embedded into $\ell_1$ with constant distortion. However, the exponential dependency on $k$ in their distortion bound remains the state of the art. Their embedding is obtained by a so-called onion-peeling approach: removing one layer of the graph at a time at the cost of incurring $O(1)$ distortion multiplicatively, and recursively embedding the resulting $(k-1)$-outerplanar graph. In this paper, we devise a new onion peeling approach that peels $k/2$ layers off the input $k$-outerplanar graph at every step. As a result, we obtain the first embedding of $k$-outerplanar graphs into trees and $\ell_1$ with distortion $\operatorname{poly}(k)$, an exponential improvement over the best-known distortion bound [SODA'03]. Our result of embeddings into trees comes closer to the distortion lower bound $Ω(k)$ for $k$-outerplanar graphs.
Comments22 pages, 2 figures, SODA'27
