Metaheuristic Algorithms for Solving Roman {2}-Domination Problem

A Roman $\{2\}$-dominating function ($Rom2DF$) on a graph $G(V,E)$ is a function $g:V \rightarrow \{0,1,2\}$ of  $G$ such that for every vertex $x\in V$ with $g(x)=0,$ either there exists a neighbor $y$ of $x$ with $g(y)=2$ or at least two neighbors, $u,v$  with $g(u)=g(v)=1$. The value $w(g)=\sum_{x\in V}g(x)$ is the weight of the Rom2DF. The minimum weight of a Rom2DF of $G$ is called the \textit{Roman $\{2\}$-domination number} denoted by \textit{$\gamma_{\{R2\}}(G)$}. Since determining  \textit{$\gamma_{\{R2\}}(G)$} of a graph $G$ is NP-hard and no metaheuristic algorithms have been proposed for the same, two procedures based on genetic algorithm are proposed as a solution for the Roman \{2\}-domination problem. One of the proposed methods employs a random initial population, while the other uses a population generated using heuristics. Experiments have been carried out on graphs generated using \textit{Erdős–Rényi} model, a popular model for graph generation and $Harwell \;Boeing (HB)$ dataset. The experimental results demonstrate that both approaches provide a near optimal solution which is well within the known lower and upper bounds for the problem. The experimental results further show that the procedure based on random initial population has outperformed the heuristic based procedure.