A note on the random greedy independent set algorithm

TL;DR

This paper analyzes the size of independent sets formed by a random greedy algorithm in regular hypergraphs, providing lower bounds under certain conditions, with implications for combinatorics.

Contribution

It extends previous bounds on independent set sizes to hypergraphs with specific degree and codegree conditions, generalizing known results.

Findings

Independent sets have size at least Omega(N * (log N / D)^{1/(r-1)}) with high probability.

The results apply to hypergraphs satisfying certain degree and codegree constraints.

Generalizes bounds from the H-free process to broader hypergraph classes.

Abstract

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.