It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into $\ell_1$ with constant distortion. However, given an $n$-vertex weighted planar graph, the best upper bound on the distortion is only $O(\sqrt{\log n})$, by Rao [SoCG99]. In this paper we study the case where there is a set $K$ of terminals, and the goal is to embed only the terminals into $\ell_1$ with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into $\ell_1$. The more general case, where the set of terminals can be covered by $\gamma$ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of $O(\log \gamma)$ by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to $O(\sqrt{\log\gamma})$. Since every planar graph has at most $O(n)$ faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into $\ell_1$. Therefore, our result provides a polynomial time $O(\sqrt{\log \gamma})$-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by $\gamma$ faces.

翻译：Gupta 等人 [Combinatorica04] 曾猜想每个平面图都能以常数失真嵌入 $\ell_1$ 空间。然而，对于给定的 $n$ 顶点加权平面图，目前最佳失真上界仅为 $O(\sqrt{\log n})$（由 Rao [SoCG99] 提出）。本文研究存在终端集 $K$ 且目标仅为将该终端集以低失真嵌入 $\ell_1$ 的情形。在开创性工作中，Okamura 与 Seymour [J.Comb.Theory81] 证明：若所有终端位于同一面，则可等距嵌入 $\ell_1$。更一般的情形——终端集可被 $\gamma$ 个面覆盖——由 Lee 与 Sidiropoulos [STOC09] 及 Chekuri 等人 [J.Comb.Theory13] 研究。当前最优结果为 Krauthgamer、Lee 与 Rika [SODA19] 提出的 $O(\log \gamma)$ 上界。我们的贡献在于将该上界进一步改进至 $O(\sqrt{\log \gamma})$。由于每个平面图至多包含 $O(n)$ 个面，任何对此结果的进一步改进都将成为重大突破，直接改进 Rao 长期保持的上界。此外，众所周知流-割间隙等价于 $\ell_1$ 最佳嵌入的失真。因此，我们的结果为平面图上所有需求对可被 $\gamma$ 个面覆盖的情形，提供了稀疏割问题的多项式时间 $O(\sqrt{\log \gamma})$-近似算法。