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.

[1]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[2]  Uriel Feige,et al.  Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT , 1995, Proceedings Third Israel Symposium on the Theory of Computing and Systems.

[3]  P. Delsarte.  Hahn Polynomials, Discrete Harmonics, and t-Designs , 1978 .

[4]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[5]  Ali Ridha Mahjoub,et al.  On the cut polytope , 1986, Math. Program..

[6]  David R. Karger,et al.  Approximate graph coloring by semidefinite programming , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.