Asymptotic optimality of degree-greedy discovering of independent sets in Configuration Model graphs

TL;DR

This paper analyzes the asymptotic performance of degree-greedy algorithms for finding large independent sets in large sparse random graphs, providing conditions for near-optimal solutions and insights into classical phenomena.

Contribution

It characterizes the asymptotic behavior of degree-greedy algorithms in configuration model graphs and offers new proofs and insights for Erdős-Rényi graphs.

Findings

Degree-greedy algorithms achieve asymptotically optimal independent sets under certain conditions.

Provides a new proof of the $e$-phenomenon in Erdős-Rényi graphs.

Characterizes the asymptotic independence number in large sparse graphs.

Abstract

Finding independent sets of maximum size in fixed graphs is well known to be an NP-hard task. Using scaling limits, we characterise the asymptotics of sequential degree-greedy explorations and provide sufficient conditions for this algorithm to find an independent set of asymptotically optimal size in large sparse random graphs with given degree sequences. In the special case of sparse Erd\"os-R\'enyi graphs, our results allow to give a simple proof of the so-called $e$-phenomenon identified by Karp and Sipser for matchings and to give an alternative characterisation of the asymptotic independence number.