Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the random greedy independent set algorithm

Patrick Bennett, Tom Bohman

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 $ Ω\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.

A note on the random greedy independent set algorithm

Abstract

Let be a fixed constant and let be an -uniform, -regular hypergraph on vertices. Assume further that for some . Consider the random greedy algorithm for forming an independent set in . 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 with the property that is not in the current independent set and contains no edge in ). Note that this process terminates at a maximal subset of vertices with the property that this set contains no edge of ; that is, the process terminates at a maximal independent set. We prove that if satisfies certain degree and codegree conditions then there are 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 -free process due to Bohman and Keevash and produces objects of interest in additive combinatorics.

Paper Structure

This paper contains 7 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. The $k$-AP-free process and Gowers uniformity norm
  3. Hypergraph Turán
  4. Lower bound: Proof of Theorem \ref{['theory:main']}
  5. Crude bounds
  6. Dynamic concentration
  7. Subgraph counts: Proof of Theorem \ref{['theory:count']}

Key Result

Theorem 1.1

Let $r\ge 3$ and $\epsilon > 0$ be fixed. Let ${\mathcal{H}}$ be a $r$-uniform, $D$-regular hypergraph on $N$ vertices such that $D > N^{\epsilon}$. If and $\Gamma( {\mathcal{H}} ) < D^{1- \epsilon}$ then the random greedy independent set algorithm produces an independent set $I$ in ${\mathcal{H}}$ with with probability $1 - \exp\left\{ - N^{\Omega(1)} \right\}$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 2.1
  • Lemma 2.2
  • proof
  • Corollary 3.1
  • Definition 1: Degrees of Sets
  • Definition 2: Codegrees
  • Lemma 4.1
  • proof
  • ...and 7 more