On the integrality gap of the maximum-cut semidefinite programming relaxation in fixed dimension

TL;DR

This paper introduces a convex optimization framework to precisely analyze the integrality gap of the maximum-cut semidefinite programming relaxation across fixed dimensions, providing new lower bounds.

Contribution

It formulates a factor-revealing convex optimization problem that computes the maximum ratio between rank-$n$ solutions and optimal cuts, advancing understanding of the relaxation's limitations.

Findings

Provides a method to compute lower bounds for the integrality gap.

Establishes a convex optimization problem for each fixed dimension.

Enhances understanding of the maximum-cut SDP relaxation.

Abstract

We describe a factor-revealing convex optimization problem for the integrality gap of the maximum-cut semidefinite programming relaxation: for each $n \geq 2$ we present a convex optimization problem whose optimal value is the largest possible ratio between the value of an optimal rank-$n$ solution to the relaxation and the value of an optimal cut. This problem is then used to compute lower bounds for the integrality gap.