Online Firefighting on Trees

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. We 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 the greedy algorithm on finite trees, which maximises at each turn the amount of vertices protected, is 1/2-competitive for both online versions; this was previously known only in special cases of Firefighter. We also show that, for Firefighter, the optimal competitive ratio of online algorithms ranges between 1/2 and the inverse of the golden ratio. The greedy algorithm is optimal if the number of firefighters is not bounded and we propose an optimal online algorithm which reaches the inverse of the golden ratio if at most 2 firefighters are available. Finally, we show that on infinite trees with linear growth, any firefighter sequence stronger than a non-zero periodic sequence is sufficient to contain the fire, even when revealed online