Algorithms for the global domination problem

A dominating set $D$ in a graph $G$ is a subset of its vertices such that every its vertex that does not belong to set $D$ is adjacent to at least one vertex from set $D$. A set of vertices of graph $G$ is a global dominating set if it is a dominating set for both, graph $G$ and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be $NP$-hard. We show that it remains $NP$-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.