# Saving Critical Nodes with Firefighters is FPT

## TL;DR

This paper studies the firefighting problem on graphs, focusing on saving critical nodes with limited firefighters, and establishes fixed-parameter tractability results, hardness, and kernelization bounds for different variants.

## Contribution

It introduces the first fixed-parameter tractability results for saving critical nodes in the firefighting game, connecting it with important separators and analyzing related spreading models.

## Key findings

- FPT algorithm for saving critical nodes with a bounded number of firefighters
- Hardness results for the spreading model of firefighting
- No polynomial kernelization for the problem on trees

## Abstract

We consider the problem of firefighting to save a critical subset of nodes. The firefighting game is a turn-based game played on a graph, where the fire spreads to vertices in a breadth-first manner from a source, and firefighters can be placed on yet unburnt vertices on alternate rounds to block the fire. In this work, we consider the problem of saving a critical subset of nodes from catching fire, given a total budget on the number of firefighters. We show that the problem is para-NP-hard when parameterized by the size of the critical set. We also show that it is fixed-parameter tractable on general graphs when parameterized by the number of firefighters. We also demonstrate improved running times on trees and establish that the problem is unlikely to admit a polynomial kernelization (even when restricted to trees). Our work is the first to exploit the connection between the firefighting problem and the notions of important separators and tight separator sequences. Finally, we consider the spreading model of the firefighting game, a closely related problem, and show that the problem of saving a critical set parameterized by the number of firefighters is W[2]-hard, which contrasts our FPT result for the non-spreading model.