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

TL;DR

This work analyzes the greedy independent set on ${\mathcal G}(n,c/n)$, focusing on large deviations from the typical size $s_c n$ where $s_c=(1/c)\log(1+c)$. By identifying a least-cost trajectory $\hat{y}_s(x)$ via a discrete Euler–Lagrange framework, the authors derive a simple closed-form rate function for the large-deviation principle of both the trajectory of available vertices and the final size, confirming the sharpness of Pittel's earlier bounds. The approach provides a self-contained and elementary alternative to the viscosity-solution methods used previously (Bermolen et al., 2020) and yields tail estimates applicable to other random growth and exploration processes. Overall, the paper advances understanding of tail behavior in greedy processes on sparse random graphs and offers a versatile analytical technique for similar problems.

Abstract

We study the greedy independent set algorithm on sparse Erdős-Rényi 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.

Figures (1)

• Figure 1: In both figures, $c=e$. The rate function is at left, intersecting the $s$-axis at $s_c\approx 0.483$. At right, the expected trajectory $\bar{y}_{c}$ is dotted, between two least-cost deviating trajectories $\hat{y}_s$, associated with the values $a=1/2$ ($s\approx 0.243$) and $a=2$ ($s\approx 0.704$).

Theorems & Definitions (2)

• Theorem 1
• Theorem 2