About the infinite windy firebreak location problem

Abstract

The severity of wildfires can be mitigated using preventive measures like the construction of firebreaks, which are strips of land from which the vegetation is completely removed. In this paper, we model the problem of wildfire containment as an optimization problem on infinite graphs called Infinite Windy Firebreak Location. A land of unknown size is modeled as an infinite undirected graph in which the vertices correspond to areas subject to fire and edges represent fire propagation from one area to another. A firebreak construction is modeled as removing the edge between two vertices. The number of firebreaks that can be installed depends on budget constraints. We assume that a fire ignites in a subset of vertices and propagates to the neighbors. The goal is to select a subset of edges to remove in order to contain the fire and avoid burning an infinite part of the graph. We prove that Infinite Windy Firebreak Location is coNP-complete in restricted cases, and we address some polynomial cases. We show that Infinite Windy Firebreak Location polynomially reduces to Min Cut for certain classes of graphs like infinite grid graphs and polyomino-grids.