# The integrality gap of the Goemans--Linial SDP relaxation for Sparsest Cut is at least a constant multiple of $\sqrt{\log n}$

## TL;DR

This paper establishes that the Goemans--Linial SDP relaxation for the Sparsest Cut problem has an integrality gap that grows at least proportionally to the square root of the logarithm of the number of vertices, indicating a fundamental limit of this approach.

## Contribution

It proves a lower bound of ( ext{log} n) on the integrality gap of the Goemans--Linial SDP relaxation for the Sparsest Cut problem, advancing understanding of approximation limits.

## Key findings

- Integrality gap is at least ( ext{log} n)
- Limits the effectiveness of the SDP relaxation for large graphs
- Provides a fundamental bound for approximation algorithms

## Abstract

We prove that the integrality gap of the Goemans--Linial semidefinite programming relaxation for the Sparsest Cut Problem is $\Omega(\sqrt{\log n})$ on inputs with $n$ vertices.