A locating-dominating set $D$ of a graph $G$ is a dominating set of $G$ where each vertex not in $D$ has a unique neighborhood in $D$, and the Locating-Dominating Set problem asks if $G$ contains such a dominating set of bounded size. This problem is known to be $\mathsf{NP-hard}$ even on restricted graph classes, such as interval graphs, split graphs, and planar bipartite subcubic graphs. On the other hand, it is known to be solvable in polynomial time for some graph classes, such as trees and, more generally, graphs of bounded cliquewidth. While these results have numerous implications on the parameterized complexity of the problem, little is known in terms of kernelization under structural parameterizations. In this work, we begin filling this gap in the literature. Our first result shows that Locating-Dominating Set, when parameterized by the solution size $d$, admits no $2^{o(d \log d)}$ time algorithm unless the Exponential Time Hypothesis fails; as a corollary, we also show that no $n^{o(d)}$ time algorithm exists under ETH, implying that the naive $\mathsf{XP}$ algorithm is essentially optimal. We present an exponential kernel for the distance to cluster parameterization and show that, unless $\mathsf{NP-hard} \subseteq \mathsf{NP-hard}/$\mathsf{poly}$, no polynomial kernel exists for Locating-Dominating Set when parameterized by vertex cover nor when parameterized by distance to clique. We then turn our attention to parameters not bounded by neither of the previous two, and exhibit a linear kernel when parameterizing by the max leaf number; in this context, we leave the parameterization by feedback edge set as the primary open problem in our study.

翻译：定位支配集 $D$ 是图 $G$ 的一个支配集，其中不在 $D$ 中的每个顶点在 $D$ 中具有唯一的邻域，而定位支配集问题询问 $G$ 是否包含大小有界的此类支配集。该问题已知在限制图类（如区间图、分裂图及平面二部次立方图）上是 $\mathsf{NP-hard}$ 的。另一方面，在某些图类（如树以及更一般的团宽有界图）上，该问题可在多项式时间内求解。尽管这些结果对问题的参数化复杂度有多种影响，但在结构参数化下的核化方面知之甚少。在本工作中，我们开始填补这一文献空白。我们的第一个结果表明：当以解大小 $d$ 为参数时，定位支配集问题不存在 $2^{o(d \log d)}$ 时间算法，除非指数时间假说（ETH）不成立；作为推论，我们还证明在 ETH 下不存在 $n^{o(d)}$ 时间算法，这意味着朴素的 $\mathsf{XP}$ 算法本质上是最优的。我们针对聚类距离参数化给出了指数核，并表明除非 $\mathsf{NP-hard} \subseteq \mathsf{NP-hard}/$\mathsf{poly}$，否则当以顶点覆盖或以到团的距离为参数时，定位支配集问题不存在多项式核。随后，我们转向不受前述两个参数限制的参数，并在以最大叶数为参数时给出了线性核；在此背景下，我们将反馈边集参数化留作本研究的主要开放问题。