Finding Minimum Distance Preservers: A Parameterized Study

2026-03-22

AI Summary

This study investigates the computational complexity of finding a minimum distance-preserving subgraph that maintains shortest paths between specified sets of vertices in a given graph. Focusing on the Subsetwise Distance Preserving (SDP) and Pairwise Distance Preserving (PDP) problems, the work presents the first systematic parameterized complexity analysis with respect to parameters such as the number of terminals, vertex cover number, and treewidth. Through NP-hardness proofs, W[1]-hardness reductions, and the design of fixed-parameter tractable (FPT) algorithms, the authors uncover fundamental differences between SDP and PDP across graph classes: both problems are NP-hard and W[1]-hard on grid subgraphs; both become FPT with respect to the number of terminals on full grids; SDP is FPT parameterized by vertex cover number, whereas PDP remains NP-hard even when the vertex cover number is as small as three.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT
• Web Mining and Content Analysis: Topic discovery and tracking

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}

Problem

• distance preserver
• computational complexity
• parameterized complexity
• shortest paths
• graph algorithms

Innovation

• distance preserver
• parameterized complexity
• vertex cover
• treewidth
• fixed-parameter tractability