Large Deviation Principle for the Greedy Exploration Algorithm over Erd\"os-R\'enyi Graphs

TL;DR

This paper establishes a large deviation principle for a greedy exploration algorithm on Erd"os-Rényi graphs, providing explicit formulas for the rate function and insights into the size of the maximum independent set.

Contribution

It introduces a large deviation principle for the greedy exploration process on ER graphs, with explicit rate functions and analysis of independent set sizes.

Findings

Large deviation principle proven for the exploration process

Explicit formula for the rate function derived

Probability bounds for maximum independent set size analyzed

Abstract

We prove a large deviation principle for a greedy exploration process on an Erd\"os-R\'enyi (ER) graph when the number of nodes goes to infinity. To prove our main result, we use the general strategy to study large deviations of processes proposed by Feng and Kurtz, based on the convergence of non-linear semigroups. The rate function can be expressed in a closed-form formula, and associated optimization problems can be solved explicitly, providing the large deviation trajectory. Also, we derive an LDP for the size of the maximum independent set discovered by such an algorithm and analyze the probability that it exceeds known bounds for the maximal independent set. We also analyze the link between these results and the landscape complexity of the independent set and the exploration dynamic.