I had thought that the Goemans-Williamson approximation algorithm was the best for MAXCUT. To quote from Wikipedia:
The polynomial-time approximation algorithm for Max-Cut with the best known approximation ratio is a method by Goemans and Williamson using semidefinite programming and randomized rounding that achieves an approximation ratio $α \approx 0.878$.
However, I came across a recent paper that claims the following:
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.
I am trying to understand how significant the results of this paper are and put them in context. Does the algorithm that the authors propose do better than Goemans and Williamson for a large class of graphs? If so, is this a breakthrough result?