In a simple connected graph $G=(V,E)$, a subset of vertices $S \subseteq V$ is a dominating set if any vertex $v \in V\setminus S$ is adjacent to some vertex $x$ from this subset. A number of real-life problems can be modeled using this problem which is known to be among the difficult NP-hard problems in its class. We formulate the problem as an integer liner program (ILP) and compare the performance with the two earlier existing exact state-of-the-art algorithms and exact implicit enumeration and heuristic algorithms that we propose here. Our exact algorithm was able to find optimal solutions much faster than ILP and the above two exact algorithms for middle-dense instances. For graphs with a considerable size, our heuristic algorithm was much faster than both, ILP and our exact algorithm. It found an optimal solution for more than half of the tested instances, whereas it improved the earlier known state-of-the-art solutions for almost all the tested benchmark instances. Among the instances where the optimum was not found, it gave an average approximation error of $1.18$.

翻译：在简单连通图 $G=(V,E)$ 中，顶点子集 $S \subseteq V$ 称为支配集，当且仅当任意顶点 $v \in V\setminus S$ 均与该子集中的某个顶点 $x$ 相邻。许多实际问题可借助该问题建模，而该问题属于其类别中公认的困难NP难问题之一。我们将此问题表述为整数线性规划（ILP），并将其性能与先前已有的两种精确状态最优算法，以及本文提出的精确隐式枚举算法和启发式算法进行了比较。对于中等稠密度的实例，我们的精确算法能够比ILP及上述两种精确算法更快地找到最优解。对于规模较大的图，我们的启发式算法在速度上远超ILP和精确算法，并在超过半数的测试实例中找到了最优解；同时，该算法在几乎所有测试基准实例上改进了先前已知的状态最优解。在未找到最优解的实例中，其平均近似误差为 $1.18$。