Metric Dimension and Geodetic Set Parameterized by Vertex Cover

For a graph G, a subset S ⊆ V(G) is called a resolving set of G if, for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ 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 ⊆ V(G) of size at most k such that, for any vertex u ∈ V(G), there are two vertices s₁, s₂ ∈ S such that u lies on a shortest path from s₁ to s₂. These two classical problems are known 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. We observe that both problems admit an FPT algorithm running in 2^𝒪(vc²) ⋅ n^𝒪(1) time, and a kernelization algorithm that outputs a kernel with 2^𝒪(vc) vertices, where vc is the vertex cover number. We prove that unless the Exponential Time Hypothesis (ETH) fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, do not admit - an FPT algorithm running in 2^o(vc²) ⋅ n^𝒪(1) time, nor - a kernelization algorithm that does not increase the solution size and outputs a kernel with 2^o(vc) vertices. We only know of one other problem in the literature that admits such a tight algorithmic lower bound with respect to vc. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.