Reducing Dominating Sets in Graphs

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.