Graph Protection under Multiple Simultaneous Attacks: A Heuristic Approach

TL;DR

A variable neighborhood search algorithm is proposed in which the feasibility of the solution is checked by introducing the concept of quasi-feasibility, which is realized by careful sampling within the set of all possible attacks.

Abstract

This work focuses on developing an effective meta-heuristic approach to protect against simultaneous attacks on nodes of a network modeled using a graph. Specifically, we focus on the $k$-strong Roman domination problem, a generalization of the well-known Roman domination problem on graphs. This general problem is about assigning integer weights to nodes that represent the number of field armies stationed at each node in order to satisfy the protection constraints while minimizing the total weights. These constraints concern the protection of a graph against any simultaneous attack consisting of $k \in \mathbb{N}$ nodes. An attack is considered repelled if each node labeled 0 can be defended by borrowing an army from one of its neighboring nodes, ensuring that the neighbor retains at least one army for self-defense. The $k$-SRD problem has practical applications in various areas, such as developing counter-terrorism strategies or managing supply chain disruptions. The solution to this problem is notoriously difficult to find, as even checking the feasibility of the proposed solution requires an exponential number of steps. We propose a variable neighborhood search algorithm in which the feasibility of the solution is checked by introducing the concept of quasi-feasibility, which is realized by careful sampling within the set of all possible attacks. Extensive experimental evaluations show the scalability and robustness of the proposed approach compared to the two exact approaches from the literature. Experiments are conducted with random networks from the literature and newly introduced random wireless networks as well as with real-world networks. A practical application scenario, using real-world networks, involves applying our approach to graphs extracted from GeoJSON files containing geographic features of hundreds of cities or larger regions.

Figures (10)

• Figure 1: 3-SRD problem: An attack $P=\{A,B, E\}$ (filled in gray) can be defended by a function $f$ provided with $f(D)=f(A)= f(B)=0$, $f(C)=3$ and $f(E)=1$ with a $\omega(f) =4$. But, the attack $P=\{A,B, D\}$ cannot.
• Figure 2: 3-SRD problem: An optimal strategy for the $3$-SRD function $f$. It comes with $\omega(f)=5$.
• Figure 3: The city of Berlin divided into districts: the Queen adjacency graph
• Figure 4: Relative average improvements of Vns over Greedy approach on benchmark set Random.
• Figure 5: Relative average improvements of Vns over Greedy approach on benchmark set Wireless.