A dominating set D in a graph G is a subset of its vertices such that every vertex of the graph which 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. The problem is known to be NP-hard. Neither exact nor approximation algorithm existed . We propose two exact solution methods, one of them being based on an integer linear program (ILP) formulation, three heuristic algorithms and a special purification procedure that further reduces the size of a global dominated set delivered by any of our heuristic algorithms. We show that the problem remains NP-hard for restricted types of graphs and specify some families of graphs for which the heuristics guarantee the optimality. The second exact algorithm turned out to be about twice faster than ILP for graphs with more than 230 vertices and up to 1080 vertices, which were the largest benchmark instances that were solved optimally. The heuristics were tested for the existing 2284 benchmark problem instances with up to 14000 vertices and delivered solutions for the largest instances in less than one minute. Remarkably, for about 52% of the 1000 instances with the obtained optimal solutions, at least one of the heuristics generated an optimal solution, where the average approximation error for the remaining instances was 1.07%.

翻译：在图G中，支配集D是顶点的一个子集，使得图中不属于D的每个顶点至少与D中的某个顶点相邻。若图G的顶点集既是图G的支配集，又是其补图的支配集，则称为全局支配集。目标在于找到基数最小的全局支配集。该问题已知为NP难问题，此前不存在精确算法或近似算法。我们提出了两种精确求解方法——其中一种基于整数线性规划（ILP）建模，三种启发式算法，以及一种专用净化程序，可进一步缩减由任一启发式算法所得的全局支配集规模。我们证明了该问题在受限图类上仍保持NP难性，并指明若干启发式算法能保证最优性的图族。对于顶点数超过230且不超过1080的最大可最优求解基准实例，第二种精确算法比ILP快约两倍。启发式算法在现有2284个顶点数达14000的基准实例上进行了测试，最大规模实例的求解时间不足一分钟。值得注意的是，在已有最优解的1000个实例中，约52%的实例至少有一种启发式算法产生了最优解，其余实例的平均近似误差为1.07%。