Asymptotic performance of the Grimmett-McDiarmid heuristic

TL;DR

This paper analyzes the asymptotic distribution of the size of stable sets found by the Grimmett-McDiarmid heuristic in random graphs, extending previous results on its typical size.

Contribution

It provides the first detailed asymptotic distribution analysis of the heuristic's output in Erdős–Rényi graphs.

Findings

Determines the asymptotic distribution of the stable set size

Extends understanding of heuristic performance beyond expected size

Provides probabilistic insights into the heuristic's behavior

Abstract

Grimmett and McDiarmid suggested a simple heuristic for finding stable sets in random graphs. They showed that the heuristic finds a stable set of size $\sim\log_2 n$ (with high probability) on a $G(n, 1/2)$ random graph. We determine the asymptotic distribution of the size of the stable set found by the algorithm.