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Wildfire Suppression: Complexity, Models, and Instances

Gustavo Delazeri, Marcus Ritt

Abstract

Wildfires cause major losses worldwide, and the frequency of fire-weather conditions is likely to increase in many regions. We study the allocation of suppression resources over time on a graph-based representation of a landscape to slow down fire propagation. Our contributions are theoretical and methodological. First, we prove that this problem and related variants in the literature are NP-complete, including cases without resource-timing constraints. Second, we propose a new mixed-integer programming (MIP) formulation that obtains state-of-the-art results, showing that MIP is a competitive approach contrary to earlier findings. Third, showing that existing benchmarks lack realism and difficulty, we introduce a physics-grounded instance generator based on Rothermel's surface fire spread model. We use these diverse instances to benchmark the literature, identifying the specific conditions where each algorithm succeeds or fails.

Wildfire Suppression: Complexity, Models, and Instances

Abstract

Wildfires cause major losses worldwide, and the frequency of fire-weather conditions is likely to increase in many regions. We study the allocation of suppression resources over time on a graph-based representation of a landscape to slow down fire propagation. Our contributions are theoretical and methodological. First, we prove that this problem and related variants in the literature are NP-complete, including cases without resource-timing constraints. Second, we propose a new mixed-integer programming (MIP) formulation that obtains state-of-the-art results, showing that MIP is a competitive approach contrary to earlier findings. Third, showing that existing benchmarks lack realism and difficulty, we introduce a physics-grounded instance generator based on Rothermel's surface fire spread model. We use these diverse instances to benchmark the literature, identifying the specific conditions where each algorithm succeeds or fails.

Paper Structure

This paper contains 19 sections, 5 theorems, 30 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Definition
  3. Notation
  4. Related Work
  5. Computational Complexity
  6. A new Mixed-Integer Programming Formulation
  7. A note on MIP models
  8. Rothermel's Model for Surface Fire Spread: A Primer
  9. No-wind, no-slope Rate of Spread
  10. Slope Factor
  11. Wind Factor
  12. Fuel Models
  13. A New Instance Generator
  14. Experiments
  15. Literature Instances
  16. ...and 4 more sections

Key Result

Theorem 4.1

WSP is NP-complete.

Figures (7)

  • Figure 1: WSP instance example with ignition vertex $s$ and $k=3$ resources released at times $t_1 = 2$, $t_2 = 3$, and $t_3 = 4$. With delay $\Delta=2$ and time horizon $H=5$ the right-hand side shows the optimal allocation $\Lambda=\{1\mapsto v_2, 2\mapsto v_4, 3\mapsto v_6\}$.
  • Figure 2: Rothermel's fire spread model. As the fire front advances, the unit volume $\Delta V$ is heated until it ignites.
  • Figure 3: Visualization of a discretized $3 \times 3$ landscape (left) and its graph representation (right). In the example, $N_{xy} = 3$, $N_{z} = 10$, and $d = 1$. The graph representation shows the coordinates of the vertices.
  • Figure 4: Histograms of values considering the 53 fuel models listed by Andrews/2018.
  • Figure 5: Illustration of the wind field used in the instance model. Each vector $\textbf{w}_{uv}$ represents the wind velocity between adjacent vertices $u$ and $v$, incorporating both directional and magnitude perturbations relative to the predominant wind direction. These variations are introduced using Perlin noise.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Definition 1: MVNP
  • Theorem 4.1
  • proof
  • Definition 2: WWSP
  • Theorem 4.2
  • proof
  • Definition 3: HWSP
  • Proposition 1
  • proof
  • Proposition 2
  • ...and 3 more