For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

翻译：对于图$G$，称子集$S\subseteq V(G)$为$G$的一个分辨集，若对任意两个顶点$u,v\in V(G)$，存在$w\in S$使得$d(w,u)\neq d(w,v)$。度量维数问题的输入为具有$n$个顶点的图$G$与正整数$k$，要求判定是否存在大小不超过$k$的分辨集。在另一个基于度量的图问题——测地集中，输入为图$G$与整数$k$，目标是判定是否存在大小不超过$k$的子集$S\subseteq V(G)$，使得对任意顶点$u\in V(G)$，存在两个顶点$s_1,s_2\in S$满足$u$位于$s_1$到$s_2$的某条最短路径上。这两个经典问题被证明在自然参数（即解的大小）以及多数结构参数（包括反馈顶点集数和路径宽度）下均难以处理。目前少数已知的可解结果指出，两者关于顶点覆盖数$vc$均为FPT。具体而言，我们观察到这两个问题均存在运行时间为$2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$的FPT算法，以及能输出包含$2^{\mathcal{O}(vc)}$个顶点的核的核化算法。我们证明除非指数时间假设不成立，否则即使在有界直径的图上，度量维数与测地集问题既不存在运行时间为$2^{o(vc^2)}\cdot n^{\mathcal{O}(1)}$的FPT算法，也不存在能缩减解大小并输出包含$2^{o(vc)}$个顶点的核的核化算法。我们技术方法的通用性使其可同时适用于这两个问题。据我们所知，文献中仅有另一个问题具有如此严格的下界。类似地，在核化实例顶点数上具有指数下界的已知问题清单也极为简短。