A graph theoretical approach to the firebreak locating problem

TL;DR

This paper models wildfire spread using a graph-theoretic approach to optimize firebreak placement, addressing the problem's complexity and providing efficient solutions for specific cases.

Contribution

It introduces the Windy Firebreak Location problem, analyzes its computational complexity, and offers a polynomial-time algorithm for certain tree instances.

Findings

The problem is NP-hard even on planar bipartite graphs with degree four.

A polynomial-time algorithm exists for tree instances.

The model incorporates probabilistic fire spread and wind effects.

Abstract

In the last decade, wildfires have become wider and more destructive. The climate change and the growth of urban areas may further increase the probability of incidence of large-scale fires. The risk of fire can be lowered with preventive measures. Among them, firefighting lines are used to stop the fire from spreading beyond them. Due to high costs of installation and maintenance, their placement must be carefully planned. In this work, we address the wildfire management problem from a theoretical point of view and define a risk function to model the fire diffusion phenomena. The land is modeled by a mixed graph in which vertices are areas subject to fire with a certain probability while edges model the probability of fire spreading from one area to another. To reduce the risk, we introduce the {\sc Windy Firebreak Location} problem that addresses the optimal positioning of firefighting lines under budget constraints. We study the complexity of the problem and prove its hardness even when the graph is planar, bipartite, with maximum degree four and the propagation probabilities are equal to one. We also show an efficient polynomial time algorithm for particular instances on trees.