# Firefighting on Trees

## TL;DR

This paper investigates the Firefighter problem on trees, introducing online variants and analyzing competitive algorithms, while also exploring conditions for fire containment based on firefighter sequences and tree structure.

## Contribution

It introduces online versions of Firefighter and Fractional Firefighter, analyzes competitive ratios, and studies conditions for fire containment related to firefighter sequences and tree growth.

## Key findings

- Greedy algorithm is 1/2-competitive on finite trees.
- Optimal online Firefighter ratio is between 1/2 and the inverse of the golden ratio.
- Sufficient conditions for fire containment depend on asymptotic comparison of firefighters and tree levels.

## Abstract

In the Firefighter problem, introduced by Hartnell in 1995, a fire spreads through a graph while a player chooses which vertices to protect in order to contain it. In this paper, we focus on the case of trees and we consider as well the Fractional Firefighter game where the amount of protection allocated to a vertex lies between 0 and 1. While most of the work in this area deals with a constant amount of firefighters available at each turn, we consider three research questions which arise when including the sequence of firefighters as part of the instance. We first introduce the online version of both Firefighter and Fractional Firefighter, in which the number of firefighters available at each turn is revealed over time. We show that a greedy algorithm on finite trees is 1/2-competitive for both online versions, which generalises a result previously known for special cases of Firefighter. We also show that the optimal competitive ratio of online Firefighter ranges between 1/2 and the inverse of the golden ratio. Next, given two firefighter sequences, we discuss sufficient conditions for the existence of an infinite tree that separates them, in the sense that the fire can be contained with one sequence but not with the other. To this aim, we study a new purely numerical game called targeting game. Finally, we give sufficient conditions for the fire to be contained, expressed as the asymptotic comparison of the number of firefighters and the size of the tree levels.