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The Firefighter problem with dynamic defence costs.

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The Cost Function Firefighter Problem introduces costs, making it harder than the classic version. However, it remains tractable on specific graph types and is proven fixed-parameter tractable for broader applications.

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Area of Science:

  • Graph theory
  • Computational complexity
  • Game theory

Background:

  • The classic Firefighter Problem models contagion spread on graphs, aiming to protect vertices.
  • It is computationally hard but solvable on restricted graph classes.

Purpose of the Study:

  • Introduce and analyze The Cost Function Firefighter Problem, a variant with time- and state-dependent vertex defense costs.
  • Investigate its computational complexity and tractability on various graph classes.

Main Methods:

  • Define the Cost Function Firefighter Problem and analyze its complexity.
  • Utilize monadic second-order logic to prove fixed-parameter tractability.
  • Conduct empirical investigations comparing different heuristic strategies.

Main Results:

  • The Cost Function problem is computationally hard, even on trees where the classic problem is tractable.
  • It is tractable on complete graphs, graphs with bounded path length, and certain trees.
  • Proved fixed-parameter tractability with respect to treewidth, budget, and maximum time step.
  • Empirical results show heuristic effectiveness varies with graph structure; state-based strategies outperform degree-based ones.

Conclusions:

  • The Cost Function Firefighter Problem presents new computational challenges but is manageable for specific parameters and graph structures.
  • Heuristic choice is critical and depends on the specific graph and cost functions involved in contagion modeling.