发表机构
Grainger College of Engineering, University of Illinois; University of California, Santa Cruz（伊利诺伊大学厄巴纳-香槟分校格拉英工程学院; 加州大学圣克鲁兹分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Steiner点移除问题，利用低度高围长扩展图构造，通过更精细的扩展性分析，将可实现拉伸的下界从$\Omega(\sqrt{\log|T|/\log\log|T|})$改进至$\Omega(\sqrt{\log|T|})$。
AI 中文摘要
在Steiner点移除问题中，给定一个图$G=(V,E)$，带有边长度函数$\ell_G: E\rightarrow \mathbb{R}_+$，以及一个终端子集$T\subseteq V$。目标是找到$G$的一个以$T$为顶点集的子式$H=(T, E_H)$，使得由$G$在$H$的边上导出的最短路径度量在小的乘法拉伸因子内保持每对终端之间的距离。Filtser证明了可以实现$O(\log |T|)$的拉伸（在多项式时间内），而Chen和Tan最近证明了可实现拉伸的下界为$\Omega\left(\sqrt{\frac{\log |T|}{\log\log |T|}}\right)$。他们的下界通过一个涉及低度高围长图的简单构造获得。这类图的存在性由低度高围长扩展图的存在性保证。在这项工作中，我们使用Chen和Tan的相同简单构造，但通过更仔细的分析利用扩展性质，将下界改进为$\Omega(\sqrt{\log |T|})$。
英文摘要
In the Steiner Point Removal problem, we are given a graph $G=(V,E)$ with an edge-length function $\ell_G: E\rightarrow \mathbb{R}_+$ and a subset $T\subseteq V$ of terminals. The goal is to find a minor $H=(T, E_H)$ of $G$ on vertex set $T$ such that the shortest path metric derived from $G$ on the edges of $H$ preserves the distance between every pair of terminals within a small multiplicative stretch. Filtser proved that a stretch of $O(\log |T|)$ can be achieved (in polynomial time), while Chen and Tan more recently proved a lower bound of $Ω\left(\sqrt{\frac{\log |T|}{\log\log |T|}}\right)$ on the achievable stretch. Their lower bound is via a simple construction involving low-degree high-girth graphs. The existence of such graphs is guaranteed through the existence of low-degree high-girth expanders. In this work, we improve the lower bound to $Ω(\sqrt{\log |T|})$ using the same simple construction of Chen and Tan but with a more careful analysis that exploits the expansion property.
