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

In this note, we describe a ${\alpha }_{GW}+\tilde{\mathrm{\Omega }}(1\mathrm{/}{d}^2)$-factor approximation algorithm for Max-Cut on weighted graphs of degree $\le d$. 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 Florén. 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.