Meta-heuristic Algorithms for Double Roman Domination Problem

A variety of domination concepts have been defined to provide better routing and defense strategies under different constraints. A double Roman dominating function (DROMDF) on a simple, undirected graph.. is a function g : V -> {0, 1, 2, 3} such that every vertex x is an element of V with g(x) = 0 is adjacent to at least two vertices y(1), y(2) with g(y(1)) = g(y(2)) = 2 or a vertex z(1) with g(z(1)) = 3. Also, a vertex p with g(p) = 1 is adjacent to at least one vertex q(1) with g(q(1)) >= 2. gamma(dR)(G), the double Roman domination number of G, is the smallest possible weight of all possible DROMDFs of G. Determining double Roman domination number of a graph is known to be NP-hard. Hence in this paper, we propose a genetic algorithm based approach for solving double Roman domination problem in which three heuristic algorithms have been proposed and problem specific crossover operator and a feasibility function has been developed. Further, we propose an ant colony optimization algorithm to solve double Roman domination problem. This paper provides an in-depth illustration of two algorithms for solving double Roman domination problem. Effectiveness of the proposed meta-heuristic algorithms is tested on the random graphs generated using NetworkX Erdos-Renyi model, a popular model for graph generation and Harwell-Boeing dataset, a well-known dataset for graph related problems. Further, we compare the results of both the meta-heuristic algorithms and the experimental results show that the proposed meta-heuristic algorithms for solving double Roman domination problem give a near optimal solution in reasonable time. Experimental results also show that the proposed ant colony optimization algorithm for solving double Roman domination problem outperforms genetic algorithm based procedure.