A dominating set of a graph $G=(V,E)$ is a subset of vertices $S\subseteq V$ such that every vertex $v\in V\setminus S$ has at least one neighbor in set $S$. The corresponding optimization problem is known to be NP-hard. The best known polynomial time approximation algorithm for the problem separates the solution process in two stages applying first a fast greedy algorithm to obtain an initial dominating set, and then it uses an iterative procedure to reduce (purify) this dominating set. The purification stage turned out to be practically efficient. Here we further strengthen the purification stage presenting four new purification algorithms. All four purification procedures outperform the earlier purification procedure. The algorithms were tested for over 1300 benchmark problem instances. Compared to the known upper bounds, the obtained solutions were about 7 times better. Remarkably, for the 500 benchmark instances for which the optimum is known, the optimal solutions were obtained for 46.33\% of the tested instances, whereas the average error for the remaining instances was about 1.01.

翻译：图$G=(V,E)$的支配集是指顶点子集$S\subseteq V$，使得每个顶点$v\in V\setminus S$在集合$S$中至少有一个邻点。该优化问题已知为NP难问题。目前已知的最佳多项式时间近似算法将求解过程分为两个阶段：首先应用快速贪心算法获得初始支配集，然后通过迭代过程约简（纯化）该支配集。实践证明纯化阶段具有实际有效性。本文进一步强化纯化阶段，提出四种新型纯化算法。所有四种纯化方法均优于先前的纯化过程。该算法在超过1300个基准问题实例上进行了测试。与已知上界相比，所得解约改进7倍。值得注意的是，在已知最优解的500个基准实例中，46.33%的测试实例获得了最优解，其余实例的平均误差约为1.01。