Probabilistic Analysis of Rule 2

TL;DR

This paper analyzes the asymptotic performance of Rule 2, a localized approximation algorithm for connected dominating sets, on random unit disk graphs, providing bounds on the expected size of the dominating set.

Contribution

It studies the probabilistic behavior of Rule 2 on random geometric graphs, offering bounds on the dominating set size and highlighting potential for improved bounds.

Findings

Expected dominating set size is O(n/(loglog n)^{3/2}) below connectivity threshold

Rule 2's performance is characterized asymptotically on random unit disk graphs

Conjecture that the current bound can be improved

Abstract

Li and Wu proposed Rule 2, a localized approximation algorithm that attempts to find a small connected dominating set in a graph. Here we study the asymptotic performance of Rule 2 on random unit disk graphs formed from n random points in an s_n by s_n square region of the plane. If s_n is below the threshold for connectivity, then Rule 2 produces a dominating set whose expected size is O(n/(loglog n)^{3/2}). We conjecture that this bound is not optimal.