A natural barrier in random greedy hypergraph matching

TL;DR

This paper analyzes the random greedy algorithm for hypergraph matching, revealing a natural barrier in the process where the unmatched vertices proportion is bounded by a specific ratio involving degrees.

Contribution

It establishes a probabilistic bound on the unmatched vertices in the hypergraph matching process, highlighting a fundamental barrier in the random greedy approach.

Findings

Unmatched vertices proportion is at most (L/D)^{1/(2(r-1)) + o(1)} with high probability.

Identifies a natural barrier in the analysis of the random greedy hypergraph matching.

Provides probabilistic bounds under specific regularity and degree conditions.

Abstract

Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most $L$ where $L \ = D/ (\log N)^{\omega(1)}$. We consider the random greedy algorithm for forming a matching in $ \mathcal{H}$. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\mathcal H}$ that are not saturated by the final matching is at most $ (L/D)^{ \frac{ 1}{ 2(r-1) } + o(1) } $. This point is a natural barrier in the analysis of the random greedy hypergraph matching process.