Complexity of Firefighting on Graphs

TL;DR

This paper studies a firefighting game on graphs, providing bounds on the minimum firefighters needed, proving NP-hardness of related decision problems, and exploring strategy complexities.

Contribution

It introduces bounds for firefighting on binary trees, proves NP-hardness of firefighter number decision, and connects results to the Hunter and Rabbit game.

Findings

Bounds for firefighter number on complete binary trees

Deciding if firefighter number <= m is NP-hard

Shortest strategies can be superpolynomial in length

Abstract

We consider a pursuit-evasion game that describes the process of extinguishing a fire burning on the nodes of an undirected graph. We denote the minimum number of firefighters required by ffn(G) and provide almost sharp bounds to this graph parameter for complete binary trees. We show that deciding whether ffn(G) <= m for given G and m is NP-hard. Furthermore, we show that shortest strategies can have superpolynomial length, leaving open whether the problem is in NP. We provide a construction that allows for transferring these results to a well-established Cops and Robbers variant called the "Hunter and Rabbit game".