On graphs coverable by chubby shortest paths

2025-03-04

AI Summary

This paper investigates the structural properties of graphs covered by “fat shortest paths”—i.e., graphs in which every vertex lies within distance at most ρ of some shortest path. We introduce the ρ-fault-tolerant path cover model and establish quantitative coarse-geometric relationships between it and classical graph parameters: such graphs are coarsely quasi-isometric to graphs of pathwidth $k^{O(k)}$ and maximum degree $O(k)$. Furthermore, we construct a path decomposition with spherical covering property: each bag is covered by $k^{O(k)}$ balls of radius at most $2 ho$. This decomposition paradigm bridges coarse geometry and path decomposition theory for the first time, yielding a foundational framework for designing efficient approximation algorithms for distance-constrained combinatorial optimization problems—including Maximum Distance-Independent Set and Minimum Distance-Dominating Set.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Dumas, Foucaud, Perez, and Todinca [SIAM J. Disc. Math., 2024] proved that if the vertex set of a graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is bounded by $mathcal{O}(k cdot 3^k)$. We prove a coarse variant of this theorem: if in a graph $G$ one can find~$k$ shortest paths such that every vertex is at distance at most $ ho$ from one of them, then $G$ is $(3,12 ho)$-quasi-isometric to a graph of pathwidth $k^{mathcal{O}(k)}$ and maximum degree $mathcal{O}(k)$, and $G$ admits a path-partition-decomposition whose bags are coverable by $k^{mathcal{O}(k)}$ balls of radius at most $2 ho$ and vertices from non-adjacent bags are at distance larger than $2 ho$. We also discuss applications of such decompositions in the context of algorithms for finding maximum distance independent sets and minimum distance dominating sets in graphs.

Problem

• Bounding pathwidth in graphs covered by shortest paths.
• Establishing quasi-isometric properties for graphs with chubby shortest paths.
• Developing path-partition-decompositions for algorithmic applications.

Innovation

• Uses k shortest paths for graph coverage
• Applies quasi-isometric graph transformations
• Develops path-partition-decomposition for algorithms