How Good is the Goemans--Williamson MAX CUT Algorithm?

The celebrated semidefinite programming algorithm for MAX CUT introduced by Goemans and Williamson was known to have a performance ratio of at least α= 2 π min0 < θ≤ π θ 1-cos θ (0.87856 < α < 0.87857); the exact performance ratio was unknown. We prove that the performance ratio of their algorithm is exactly α. Furthermore, we show that it is impossible to add valid linear constraints to improve the performance ratio.