Randomized greedy algorithm for independent sets in regular uniform hypergraphs with large girth

TL;DR

This paper analyzes a randomized greedy algorithm for finding large independent sets in regular hypergraphs with large girth, extending previous graph results to hypergraphs and proving concentration of the independent set size.

Contribution

It extends earlier graph results to hypergraphs, providing bounds on independent set sizes and proving concentration for linear hypergraphs with bounded degree.

Findings

Expected independent set size approaches a bound as girth increases.

The size of independent sets concentrates around the mean in linear hypergraphs.

The analysis generalizes known results from graphs to hypergraphs.

Abstract

In this paper, we consider a randomized greedy algorithm for independent sets in $r$-uniform $d$-regular hypergraphs $G$ on $n$ vertices with girth $g$. By analyzing the expected size of the independent sets generated by this algorithm, we show that $\alpha(G)\geq (f(d,r)-\epsilon(g,d,r))n$, where $\epsilon(g,d,r)$ converges to $0$ as $g\rightarrow\infty$ for fixed $d$ and $r$, and $f(d,r)$ is determined by a differential equation. This extends earlier results of Gamarnik and Goldberg for graphs. We also prove that when applying this algorithm to uniform linear hypergraphs with bounded degree, the size of the independent sets generated by this algorithm concentrate around the mean asymptotically almost surely.