Making Life More Confusing for Firefighters

TL;DR

This paper introduces the Temporal Firefighter problem, extending the classic Firefighter problem to temporal graphs, demonstrating its NP-Completeness, and providing an FPT algorithm based on a specific temporal graph parameter.

Contribution

It models the impact of civilian aid on firefighting strategies using temporal graphs and proves the problem's NP-Completeness, offering a fixed-parameter tractable algorithm.

Findings

Temporal Firefighter is NP-Complete on most graph classes.

Civilian aid complicates firefighting strategies.

An FPT algorithm is developed for the problem.

Abstract

It is well known that fighting a fire is a hard task. The Firefighter problem asks how to optimally deploy firefighters to defend the vertices of a graph from a fire. This problem is NP-Complete on all but a few classes of graphs. Thankfully, firefighters do not have to work alone, and are often aided by the efforts of good natured civilians who slow the spread of a fire by maintaining firebreaks when they are able. We will show that this help, although well-intentioned, unfortunately makes the optimal deployment of firefighters an even harder problem. To model this scenario we introduce the Temporal Firefighter problem, an extension of Firefighter to temporal graphs. We show that Temporal Firefighter is also NP-Complete, and remains so on all but one of the underlying classes of graphs on which Firefighter is known to have polynomial time solutions. This motivates us to explore making use of the temporal structure of the graph in our search for tractability, and we conclude by presenting an FPT algorithm for Temporal Firefighter with respect to the temporal graph parameter vertex-interval-membership-width.