# Central limit theorems for combinatorial optimization problems on sparse Erd\H{o}s-R\'enyi graphs

## TL;DR

This paper develops a new method to prove central limit theorems for certain combinatorial optimization problems on sparse Erdős-Rényi graphs, advancing understanding of their probabilistic fluctuations.

## Contribution

It introduces a novel approach to establish CLTs for problems exhibiting endogeny or long-range independence on sparse graphs.

## Key findings

- Applicable to maximum weight matching and edge cover problems
- Demonstrates CLTs for problems with replica symmetry
- Advances probabilistic analysis in sparse graph optimization

## Abstract

For random combinatorial optimization problems, there has been much progress in establishing laws of large numbers and computing limiting constants for the optimal value of various problems. However, there has not been as much success in proving central limit theorems. This paper introduces a method for establishing central limit theorems in the sparse graph setting. It works for problems which display a key property which has been variously called "endogeny", "long-range independence", and "replica symmetry" in the literature. Examples of such problems are maximum weight matching, $\lambda$-diluted minimum matching, and optimal edge cover.