Finding Minimum Distance Preservers: A Parameterized Study

Abstract: 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}