Fixed Parameter Tractable Algorithm for Firefighting Problem

TL;DR

This paper introduces fixed parameter tractable algorithms for the firefighter problem, demonstrating its solvability under various parameters on different graph classes, including weighted and multi-source scenarios.

Contribution

It presents new FPT algorithms for the firefighter problem parameterized by burnt vertices, protected vertices, and extends to weighted, multi-source, and specific graph classes.

Findings

FPT algorithm for general graphs based on burnt vertices

FPT algorithms for degree-bounded and unicyclic graphs

Extension to weighted graphs and multiple fire sources

Abstract

The firefighter problem is defined as below. A fire initially breaks out at a vertex r on a graph G. In each step, a firefighter chooses to protect one vertex, which is not yet burnt. And the fire spreads out to its unprotected neighboring vertices afterwards. The objective of the problem is to choose a sequence of vertices to protect, in order to save maximum number of vertices from the fire. In this paper, we will introduce a parameter k into the firefighter problem and give several FPT algorithms using a random separation technique of Cai, Chan and Chan. We will prove firefighter problem is FPT on general graph if we take total number of vertices burnt to be a parameter. If we parameterize the number of protected vertices, we discover several FPT algorithms of the firefighter problem on degree bounded graph and unicyclic graph. Furthermore, we also study the firefighter problem on weighted and valued graph, and the problem with multiple fire sources on degree-bounded graph.