For a given graph $G$ and a subset of vertices $S$, a \emph{distance preserver} is a subgraph of $G$ that preserves shortest paths between the vertices of $S$. We distinguish between a \emph{subsetwise} distance preserver, which preserves distances between all pairs in $S$, and a \emph{pairwise} distance preserver, which preserves distances only between specific pairs of vertices in $S$, given in the input. While a large body of work is dedicated to upper and lower bounds on the size of distance preservers and, more generally, graph spanners, the computational complexity of finding the minimum distance preserver has received comparatively little attention. We consider the respective \scup{Subsetwise Distance Preserver}\xspace (\scup{SDP}\xspace) and \scup{Pairwise Distance Preserver}\xspace (\scup{PDP}\xspace) problems and initiate the study of their computational complexity. We provide a detailed complexity landscape with respect to natural parameters, including the number of terminals, solution size, vertex cover, and treewidth. Our main contributions are as follows: \begin{itemize} \setlength{\itemsep}{0.5em} \item Both \scup{PDP}\xspace and \scup{SDP}\xspace are \nph\ even on subgraphs of the grid. Moreover, when parameterized by the number of terminals, the problems are \wh{1}\ on subgraphs of the grid, while they become \textsc{FPT}\ on full grids. \item \scup{PDP}\xspace is \nph\ on graphs of vertex cover $3$, while \scup{SDP}\xspace is \textsc{FPT}\ when parameterized by the vertex cover of the graph. Thus, the vertex cover parameter distinguishes the two variants. \item Both problems are \textsc{FPT}\ when parameterized by the number of terminals and the treewidth of the graph. \end{itemize}

翻译：对于给定图$G$和顶点子集$S$，\emph{距离保持子图}是$G$的一个子图，它保留$S$中顶点之间的最短路径。我们区分\emph{子集式}距离保持子图（保留$S$中所有顶点对之间的距离）和\emph{成对式}距离保持子图（仅保留输入中指定的$S$内特定顶点对之间的距离）。尽管大量工作致力于距离保持子图以及更一般的图扳手的大小的上下界，但寻找最小距离保持子图的计算复杂性却相对较少受到关注。我们考虑相应的\scup{子集式距离保持子图}\xspace（\scup{SDP}\xspace）和\scup{成对式距离保持子图}\xspace（\scup{PDP}\xspace）问题，并开始对其计算复杂性进行研究。我们根据自然参数（包括端点数、解的大小、顶点覆盖和树宽）提供了一个详细的复杂性景观。我们的主要贡献如下：\begin{itemize} \setlength{\itemsep}{0.5em} \item 即使局限于网格子图，\scup{PDP}\xspace和\scup{SDP}\xspace都是\nph\的。此外，当以端点数参数化时，这些问题在网格子图上为\wh{1}\难，而在整个网格上则变为\textsc{FPT}。\item \scup{PDP}\xspace在顶点覆盖为$3$的图中是\nph\的，而\scup{SDP}\xspace在以图的顶点覆盖为参数化时是\textsc{FPT}的。因此，顶点覆盖参数区分了这两个变体。\item 当以端点数与图的树宽参数化时，这两个问题都是\textsc{FPT}的。\end{itemize}