Parameterized Complexity of Firefighting Revisited

TL;DR

This paper investigates the computational complexity of the Firefighter problem, showing fixed-parameter tractability on general graphs when parameterized by burned vertices, and W[1]-hardness when parameterized by unburned vertices, with improved algorithms for trees.

Contribution

It completes the complexity landscape of the Firefighter problem by establishing FPT results, kernelization bounds, and hardness results for various graph classes and parameters.

Findings

FPT on general graphs with respect to burned vertices

No polynomial kernel on trees for the problem

W[1]-hardness on bipartite graphs when parameterized by unburned vertices

Abstract

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.