For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} 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 {\sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be \NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is \W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is \FPT\ when parameterized by the vertex cover number and asked about its complexity under \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc Metric Dimension} is \W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is \W[1]-hard parameterized by the pathwidth. On the positive side, we show that {\sc Metric Dimension} is \FPT\ when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

翻译：对于图$G$，子集$S \subseteq V(G)$称为一个\emph{分辨集}，如果对任意两个顶点$u,v \in V(G)$，存在顶点$w \in S$使得$d(w,u) \neq d(w,v)$。{\sc 度量维度}问题输入一个图$G$和一个正整数$k$，询问是否存在大小至多为$k$的分辨集。该问题于20世纪70年代提出，已知为\NP-难问题（见Garey与Johnson著作中的GT~61）。在参数化复杂性领域，Hartung和Nichterlein~[CCC~2013]证明该问题在自然参数$k$下为\W[2]-难。他们还观察到该问题在顶点覆盖数参数化下属于\FPT，并询问其在\emph{更小}参数（特别是反馈顶点集数）下的复杂性。我们通过证明{\sc 度量维度}问题在反馈顶点集数加路径宽度的组合参数下为\W[1]-难来回答此问题。这一结果改进了Bonnet和Purohit~[IPEC 2019]的结论，后者指出该问题在路径宽度参数化下为\W[1]-难。从正面结果看，我们证明{\sc 度量维度}问题在距离簇或距离共簇参数化下均属于\FPT，这两个参数均小于顶点覆盖数。