An explicit vector algorithm for high-girth MaxCut

TL;DR

This paper introduces an explicit vector algorithm for MaxCut that improves approximation guarantees on high-girth regular graphs, simplifying previous complex methods and outperforming existing algorithms.

Contribution

The paper presents a new explicit vector-based approximation algorithm for MaxCut on high-girth regular graphs, improving guarantees over prior classical and quantum approaches.

Findings

Better approximation guarantees for high-girth regular graphs.

Simplified algorithm compared to previous Gaussian wave process methods.

Outperforms existing classical and quantum algorithms in the specified setting.

Abstract

We give an approximation algorithm for MaxCut and provide guarantees on the average fraction of edges cut on $d$-regular graphs of girth $\geq 2k$. For every $d \geq 3$ and $k \geq 4$, our approximation guarantees are better than those of all other classical and quantum algorithms known to the authors. Our algorithm constructs an explicit vector solution to the standard semidefinite relaxation of MaxCut and applies hyperplane rounding. It may be viewed as a simplification of the previously best known technique, which approximates Gaussian wave processes on the infinite $d$-regular tree.