Minimum Eccentricity Shortest Paths in some Structured Graph Classes

TL;DR

This paper studies the problem of finding shortest paths with minimal eccentricity in structured graph classes, providing efficient algorithms for certain classes and approximation methods for others.

Contribution

It introduces linear-time algorithms for minimum eccentricity shortest paths in distance-hereditary graphs and extends polynomial-time solutions to other graph classes.

Findings

Linear-time algorithm for distance-hereditary graphs

Polynomial-time solutions for chordal and dually chordal graphs

Approximation algorithms for graphs with bounded tree-length and hyperbolicity

Abstract

We investigate the Minimum Eccentricity Shortest Path problem in some structured graph classes. It asks for a given graph to find a shortest path with minimum eccentricity. Although it is NP-hard in general graphs, we demonstrate that a minimum eccentricity shortest path can be found in linear time for distance-hereditary graphs (generalizing the previous result for trees) and give a generalised approach which allows to solve the problem in polynomial time for other graph classes. This includes chordal graphs, dually chordal graphs, graphs with bounded tree-length, and graphs with bounded hyperbolicity. Additionally, we give a simple algorithm to compute an additive approximation for graphs with bounded tree-length and graphs with bounded hyperbolicity.