The Expected Size of the Rule k Dominating Set

Abstract

Dai, Li, and Wu proposed Rule k, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. In this paper we consider the average-case performance of two closely related versions of Rule k for the model of random unit disk graphs constructed from n random points in an $\ell_n\times \ell_n$ square. We show that if $k\geq 3$ and $\ell_{n}=o(\sqrt{n}),$ then for both versions of Rule k, the expected size of the Rule k dominating set is $\Theta(\ell_{n}^{2})$ as $n\rightarrow\infty.$ It follows that, for $\ell_{n}$ in a suitable range, the expected size of the Rule k dominating sets are within a constant factor of the optimum.