A More Fine-Grained Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

TL;DR

This paper provides a detailed complexity analysis of the NP-hard problem of identifying the most vital edges in undirected graphs to increase shortest path lengths, exploring various parameters affecting computational difficulty.

Contribution

It offers refined complexity results by systematically analyzing parameters influencing tractability and hardness, advancing understanding of the problem's computational landscape.

Findings

Identified parameter regimes where the problem is tractable.

Proved hardness results for specific parameter settings.

Mapped the boundary between computationally feasible and infeasible cases.

Abstract

We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices $s$ and $t$, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest $st$-path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systematic investigation of various problem-specific parameters and their influence on the computational complexity. Charting the border between tractability and intractability, we also identify numerous challenges for future research.