  In a second seminal paper on the application of semidefinite programming to
graph partitioning problems, Goemans and Williamson showed how to formulate and
round a complex semidefinite program to give what is to date still the
best-known approximation guarantee of .836008 for Max-$3$-Cut. (This
approximation ratio was also achieved independently by De Klerk et al.) Goemans
and Williamson left open the problem of how to apply their techniques to
Max-$k$-Cut for general $k$. They point out that it does not seem
straightforward or even possible to formulate a good quality complex
semidefinite program for the general Max-$k$-Cut problem, which presents a
barrier for the further application of their techniques.
  We present a simple rounding algorithm for the standard semidefinite
programmming relaxation of Max-$k$-Cut and show that it is equivalent to the
rounding of Goemans and Williamson in the case of Max-$3$-Cut. This allows us
to transfer the elegant analysis of Goemans and Williamson for Max-3-Cut to
Max-$k$-Cut. For $k \geq 4$, the resulting approximation ratios are about $.01$
worse than the best known guarantees. Finally, we present a generalization of
our rounding algorithm and conjecture (based on computational observations)
that it matches the best-known guarantees of De Klerk et al.
