In the Steiner point removal (SPR) problem, we are given a (weighted) graph $G$ and a subset $T$ of its vertices called terminals, and the goal is to compute a (weighted) graph $H$ on $T$ that is a minor of $G$, such that the distance between every pair of terminals is preserved to within some small multiplicative factor, that is called the stretch of $H$. It has been shown that on general graphs we can achieve stretch $O(\log |T|)$ [Filtser, 2018]. On the other hand, the best-known stretch lower bound is $8$ [Chan-Xia-Konjevod-Richa, 2006], which holds even for trees. In this work, we show an improved lower bound of $\tilde\Omega\big(\sqrt{\log |T|}\big)$.

翻译：在斯坦纳点移除问题（Steiner point removal, SPR）中，给定（赋权）图 $G$ 及其顶点子集 $T$（称为终端集），目标是构造一个以 $T$ 为顶点集的（赋权）图 $H$，使得 $H$ 是 $G$ 的 minors，且任意一对终端间的距离在某个小常数因子（称为 $H$ 的拉伸因子）内保持不变。已有研究表明，在一般图上可实现拉伸因子 $O(\log |T|)$ [Filtser, 2018]。另一方面，目前已知的最佳拉伸下界为 $8$ [Chan-Xia-Konjevod-Richa, 2006]，该下界对树图同样成立。本文证明了改进的下界 $\tilde\Omega\big(\sqrt{\log |T|}\big)$。