Graph protection under multiple simultaneous attacks: A heuristic approach

This work focuses on developing a meta-heuristic approach to protect network nodes from simultaneous attacks, specifically addressing the $k$-strong Roman domination problem. The objective is to assign integer weights to the nodes, representing the number of stationed armies, to meet protection constraints while minimizing the total number of armies. A network is protected if it can repel any simultaneous attack on $k$ nodes. A node is protected if it can defend itself or if a neighboring node provides an army while retaining at least one army for self-defense. This problem formulation can be used in practical scenarios, e.g. developing counter-terrorism strategies or in coping with supply chain disruptions. The problem is difficult as even verifying the feasibility of a single solution generally requires an exponential time. Two exact approaches are proposed in the literature but applicable to small random graphs. For larger graphs, we propose an effective variable neighborhood search, where the feasibility of a solution is verified by introducing the concept of relaxed feasibility. Experiments are conducted with random networks from the literature and two introduced ad-hoc wireless and real-world networks. Extensive experimental evaluations show the robustness of the proposed approach compared to the existing approaches from the literature by significantly outperforming them in all three benchmark sets. Furthermore, we demonstrate the practical application of the proposed variable neighborhood search approach, where its solution is used to position fire stations within the city so that simultaneous fires can be extinguished efficiently while reducing the number of required fire trucks.