Finding the minimum k-weighted dominating sets using heuristic algorithms

In this work, we propose, analyze, and solve a generalization of the $k$-dominating set problem in a graph, when we consider a weighted graph. Given a graph with weights in its edges, a set of vertices is a $k$-weighted dominating set if for every vertex outside the set, the sum of the weights from it to its adjacent vertices in the set is bigger than or equal to $k$. The $k$-weighted domination number is the minimum cardinality among all $k$-weighted dominating sets. Since the problem of finding the $k$-weighted domination number is $\mathrm{NP}$-hard, we have proposed several problem-adapted construction and reconstruction techniques and embedded them in an Iterated Greedy algorithm. The resulting sixteen variants of the Iterated Greedy algorithm have been compared with an exact algorithm. Computational results show that the proposal is able to find optimal or near-optimal solutions within a short computational time. To the best of our knowledge, the $k$-weighted dominating set problem has never been studied before in the literature and, therefore, there is no other state-of-the-art algorithm to solve it. We have also included a comparison with a particular case of our problem, the minimum dominating set problem and, on average, we achieve same quality results within around 50% of computation time.