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}

Figures (2)

• Figure 1: The general overview of the construction. Terminal vertices are represented as squares, while the non-terminal vertices are represented as circles. Solid lines represent an edge while the dashed lines represent paths of length greater than one. For each color class in the BMCC instance, we create a corresponding gadget in the SDP instance. The horizontal $R_k$-row and the vertical $L_k$-column gadgets are highlighted in light blue. Each $L_i$-column gadget has two terminals $l_i^{\text{top}}$, and $l_i^{\text{bot}}$ at its endpoints, and each $R_i$-row gadget similarly has two terminals $r_i^{\text{left}}$ and $r_i^{\text{right}}$ at its endpoints. All paths between connector vertices have length $2\Delta$. Within each $L_i$-column gadget, for every $v\in L_i$, the path between $w^{\text{top}}_v$ and $w^{\text{bot}}_v$ has length $\ell$, the path between $w^{\text{top}}_v$ and $l^{\text{top}}_i$ has length $\Delta$, and the path between $w^{\text{bot}}_v$ and $l^{\text{bot}}_i$ has also length $\Delta$. Analogously, within each $R_i$-row gadget, the same length properties hold for every $u \in R_i$. Notice that endpoints of the gadgets are connected to the corresponding connector vertex via an edge. By the illustration, in $G$ it holds that $u \in R_1$, $u' \in R_k$, $v \in L_1$, and $v' \in L_k$. The gray highlighted area indicates the region where the paths in the $R_k$-row gadget and the $L_k$-column gadget intersect, which corresponds to the edges between $R_k$ and $L_k$ in the original graph $G$.
• Figure 2: Two views of the same $L_1$--$R_1$ interaction: (a) the underlying induced bipartite graph $G[L_1,R_1]$, and (b) the corresponding local intersection structure in $G'$. Note that all horizontal paths intersect with all vertical paths at their corresponding local intersection areas. As $u_1$ is connected to all the vertices of $L_1$, in $G$, its corresponding horizontal path in $G'$ shares segments of length 2 with all vertical paths in the $L_1$-column gadget. In contrast, $u_p$ is not connected to any vertex of $L_1$, so its horizontal path only intersects each vertical path at a single vertex. Vertex $u_2$ has a mixed interaction pattern, sharing segments of length 2 with vertical paths corresponding to $v_2$ and $v_p$ while only intersecting at single vertices with others. Along each vertical path, the local intersection areas are separated by paths of length $\alpha$. Moreover, every vertical and horizontal path is subdivided by $\alpha$ at the beginning (and at the end).

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Corollary 7
• Proposition 9
• Proposition 10