On the Dominating Set Problem in Random Graphs

In this paper, we study the {\sc 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 $2^{O(\log_{2}^{2}{n})}$ time. For the parameterized dominating set problem, we show that it cannot be solved in expected $O(f(k)n^{c})$ time unless the minimum dominating set problem can be approximated within a ratio of $o(\log_{2}n)$ 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 $O(f(k)n^{c})$ time when the probability $p$ depends on $n$ and equals to $\frac{1}{g(n)}$, where $g(n)< n$ is a monotonously increasing function of $n$ and its value approaches infinity when $n$ approaches infinity.