Experimental analysis of heuristic algorithms for the dominating set problem

We say a vertex upsilon in a graph G covers a vertex omega if upsilon = omega or if upsilon and omega are adjacent. A subset of vertices of G is a dominating set if it collectively covers all vertices in the graph. The dominating set problem, which is NP-hard, consists of finding a smallest possible dominating set for a graph. The straightforward greedy strategy of finding a small dominating set in a graph consists of successfully choosing vertices which cover the largest possible number of previously uncovered vertices. Several variations on this greedy heuristic are described and the results of extensive testing of these variations are presented. A more sophisticated procedure for choosing vertices, which takes into account the number of ways in which an uncovered vertex may be covered, appears to be the most successful of the algorithms which are analyzed. For our experimental testing, we used both random graphs and graphs constructed by test case generators which produce graphs with a given density and a specified size for the smallest dominating set. We found that these generators were able to produce challenging graphs for the algorithms, thus helping to discriminate among them, and allowing a greater variety of graphs to be used in the experiments.