Approximating Max-Cut on Bounded Degree Graphs: Tighter Analysis of the FKL Algorithm

TL;DR

This paper presents a refined analysis of an approximation algorithm for Max-Cut on bounded degree graphs, achieving a better approximation ratio by analyzing local improvements of the Goemans-Williamson SDP rounding.

Contribution

It provides a tighter analysis of the FKL algorithm, improving approximation guarantees for Max-Cut on graphs with degree at most d.

Findings

Achieves a $eta_{GW} + ilde{ ext{Omega}}(1/d^2)$ approximation ratio.

Improves previous bounds for unweighted graphs.

Uses a local improvement procedure on the SDP solution.

Abstract

In this note, we describe a $\alpha_{GW} + \tilde{\Omega}(1/d^2)$-factor approximation algorithm for Max-Cut on weighted graphs of degree $\leq d$. Here, $\alpha_{GW}\approx 0.878$ is the worst-case approximation ratio of the Goemans-Williamson rounding for Max-Cut. This improves on previous results for unweighted graphs by Feige, Karpinski, and Langberg and Flor\'en. Our guarantee is obtained by a tighter analysis of the solution obtained by applying a natural local improvement procedure to the Goemans-Williamson rounding of the basic SDP strengthened with triangle inequalities.