A note on the random greedy independent set algorithm

Let \(r\ge 3\) be a fixed constant and let \( {\mathcal H}\) be an \(r\)-uniform, \(D\)-regular hypergraph on \(N\) vertices. Assume further that \( D > N^\varepsilon \) for some \( \varepsilon>0 \). Consider the random greedy algorithm for forming an independent set in \( \mathcal{H}\). An in...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Bennett, Patrick, Bohman, Tom
Format: Article
Language:English
Subjects:
Algorithms
Apexes
Collection
Combinatorial analysis
Graph theory
Greedy algorithms
Lower bounds
Phase transitions
US exports
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let \(r\ge 3\) be a fixed constant and let \( {\mathcal H}\) be an \(r\)-uniform, \(D\)-regular hypergraph on \(N\) vertices. Assume further that \( D > N^\varepsilon \) for some \( \varepsilon>0 \). Consider the random greedy algorithm for forming an independent set in \( \mathcal{H}\). An independent set is chosen at random by iteratively choosing vertices at random to be in the independent set. At each step we chose a vertex uniformly at random from the collection of vertices that could be added to the independent set (i.e. the collection of vertices \(v\) with the property that \(v\) is not in the current independent set \(I\) and \( I \cup \{v\}\) contains no edge in \( \mathcal{H}\)). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of \( \mathcal{H} \); that is, the process terminates at a maximal independent set. We prove that if \( \mathcal{H}\) satisfies certain degree and codegree conditions then there are \( \Omega\left( N \cdot ( (\log N) / D )^{\frac{1}{r-1}} \right) \) vertices in the independent set produced by the random greedy algorithm with high probability. This result generalizes a lower bound on the number of steps in the \( H\)-free process due to Bohman and Keevash and produces objects of interest in additive combinatorics.
ISSN:2331-8422