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Graph Burning: Mathematical Formulations and Optimal Solutions

Author

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  • 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)

  • Lil María Xibai Rodríguez-Henríquez

    (Consejo Nacional de Ciencia y Tecnología, Mexico City 03940, Mexico
    Instituto Nacional de Astrofísica, Óptica y Electrónica, Coordinación de Ciencias Computacionales, Puebla 72840, Mexico)

  • Julio César Pérez-Sansalvador

    (Consejo Nacional de Ciencia y Tecnología, Mexico City 03940, Mexico
    Instituto Nacional de Astrofísica, Óptica y Electrónica, Coordinación de Ciencias Computacionales, Puebla 72840, Mexico)

  • Saúl Eduardo Pomares-Hernández

    (Instituto Nacional de Astrofísica, Óptica y Electrónica, Coordinación de Ciencias Computacionales, Puebla 72840, Mexico
    Laboratoire d’Analyse et d’Architecture des Aystèmes du Centre National de la Recherche Scientifique (LAAS-CNRS), Université de Toulouse, INSA, F-31400 Toulouse, France)

Abstract

The graph burning problem is an NP-hard combinatorial optimization problem that helps quantify how vulnerable a graph is to contagion. This paper introduces three mathematical formulations of the problem: an integer linear program (ILP) and two constraint satisfaction problems (CSP1 and CSP2). Thanks to off-the-shelf optimization software, these formulations can be solved optimally over arbitrary graphs; this is relevant because the only algorithms designed to date for this problem are approximation algorithms and heuristics, which do not guarantee to find optimal solutions. We empirically compared the proposed formulations using random graphs and off-the-shelf optimization software. The results show that CSP1 and CSP2 tend to reach optimal solutions in less time than the ILP. Therefore, we executed them over some benchmark graphs of order at most 5908. The previously best-known solutions for some of these graphs were improved. We draw some empirical observations from the experimental results. For instance, we find the tendency: the larger the graph’s optimal solution, the more difficult it is to find it. Finally, the resulting set of optimal solutions might be helpful as a benchmark dataset for the performance evaluation of non-exact algorithms.

Suggested Citation

  • Jesús García-Díaz & Lil María Xibai Rodríguez-Henríquez & Julio César Pérez-Sansalvador & Saúl Eduardo Pomares-Hernández, 2022. "Graph Burning: Mathematical Formulations and Optimal Solutions," Mathematics, MDPI, vol. 10(15), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2777-:d:880888
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    References listed on IDEAS

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    1. José Alejandro Cornejo Acosta & Jesús García Díaz & Ricardo Menchaca-Méndez & Rolando Menchaca-Méndez, 2020. "Solving the Capacitated Vertex K-Center Problem through the Minimum Capacitated Dominating Set Problem," Mathematics, MDPI, vol. 8(9), pages 1-16, September.
    2. Dorit S. Hochbaum & David B. Shmoys, 1985. "A Best Possible Heuristic for the k -Center Problem," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 180-184, May.
    3. Liu, Huiqing & Zhang, Ruiting & Hu, Xiaolan, 2019. "Burning number of theta graphs," Applied Mathematics and Computation, Elsevier, vol. 361(C), pages 246-257.
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