On the Dominating Set Problem in Random Graphs

In this paper, we study the Dominating Set problem in random graphs. In a random graph, each pair of vertices are joined by an edge with a probability of p, where p is a positive constant less than 1. We show that, given a random graph in n vertices, a minimum dominating set in the graph can be computed in expected 2 2 2 n) time. For the parameterized dominating set problem, we show that it cannot be solved in expected O(f(k)n) time unless the minimum dominating set problem can be approximated within a ratio of o(log2 n) in expected polynomial time, where f(k) is a function of the parameter k and c is a constant independent of n and k. In addition, we show that the parameterized dominating set problem can be solved in expected O(f(k)n) time when the probability p depends on n and equals to 1 g(n) , where g(n) < n is a monotonously increasing function of n and its value approaches infinity when n approaches infinity.