Large deviations of the greedy independent set algorithm on sparse random graphs

TL;DR

This paper analyzes the large deviations of the greedy independent set algorithm on sparse Erdős-Rényi graphs, providing a simple closed-form rate function and confirming Pittel's bounds are sharp, with implications for understanding tail behaviors in random processes.

Contribution

It introduces a straightforward method to identify the optimal trajectory for large deviations and derives a closed-form rate function, simplifying prior complex proofs.

Findings

Closed-form rate function for large deviations

Optimal trajectory characterization

Pittel's bounds are sharp

Abstract

We study the greedy independent set algorithm on sparse Erd\H{o}s-R\'enyi random graphs ${\mathcal G}(n,c/n)$. This range of $p$ is of interest due to the threshold at $c=e$, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.