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The Moving Firefighter Problem

Author

Listed:
  • Bruno R. Gutiérrez-De-La-Paz

    (Centro de Investigación en Computación del Instituto Politécnico Nacional, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

  • Jesús García-Díaz

    (Consejo Nacional de Ciencia y Tecnología, Mexico City 03940, Mexico
    Instituto Nacional de Astrofísica, Óptica y Electrónica, Coordinación de Ciencias Computacionales, Puebla 72840, Mexico
    These authors contributed equally to this work.)

  • Rolando Menchaca-Méndez

    (Centro de Investigación en Computación del Instituto Politécnico Nacional, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

  • Mauro A. Montenegro-Meza

    (Centro de Investigación en Computación del Instituto Politécnico Nacional, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

  • Ricardo Menchaca-Méndez

    (Centro de Investigación en Computación del Instituto Politécnico Nacional, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

  • Omar A. Gutiérrez-De-La-Paz

    (Centro de Investigación en Computación del Instituto Politécnico Nacional, Mexico City 07738, Mexico
    These authors contributed equally to this work.)

Abstract

The original formulation of the firefighter problem defines a discrete-time process where a fire starts at a designated subset of the vertices of a graph G . At each subsequent discrete time unit, the fire propagates from each burnt vertex to all of its neighbors unless they are defended by a firefighter that can move between any pair of vertices in a single time unit. Once a vertex is burnt or defended, it remains in that state, and the process terminates when the fire can no longer spread. In this work, we present the moving firefighter problem, which is a generalization of the firefighter problem where the time it takes a firefighter to move from a vertex u to defend vertex v is determined by a function τ . This new formulation models situations such as a wildfire or a flood, where firefighters have to physically move from their current position to the location of an entity they intend to defend. It also incorporates the notion that entities modeled by the vertices are not necessarily instantaneously defended upon the arrival of a firefighter. We present a mixed-integer quadratically constrained program (MIQCP) for the optimization version of the moving firefighter problem that minimizes the number of burnt vertices for the case of general finite graphs, an arbitrary set F ⊂ V of vertices where the fire breaks out, a single firefighter, and metric time functions τ .

Suggested Citation

  • Bruno R. Gutiérrez-De-La-Paz & Jesús García-Díaz & Rolando Menchaca-Méndez & Mauro A. Montenegro-Meza & Ricardo Menchaca-Méndez & Omar A. Gutiérrez-De-La-Paz, 2022. "The Moving Firefighter Problem," Mathematics, MDPI, vol. 11(1), pages 1-15, December.
  • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:179-:d:1019106
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    References listed on IDEAS

    as
    1. Nima Khakzad, 2018. "A Graph Theoretic Approach to Optimal Firefighting in Oil Terminals," Energies, MDPI, vol. 11(11), pages 1-14, November.
    2. O. Zambon, Mauricio J. & J. de Rezende, Pedro & C. de Souza, Cid, 2019. "Solving the geometric firefighter routing problem via integer programming," European Journal of Operational Research, Elsevier, vol. 274(3), pages 1090-1101.
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