Rounding two and three dimensional solutions of the SDP relaxation of MAX CUT

Goemans and Williamson obtained an approximation algorithm for the MAX CUT problem with a performance ratio of α_GW ≃ 0.87856. Their algorithm starts by solving a standard SDP relaxation of MAX CUT and then rounds the optimal solution obtained using a random hyperplane. In some cases, the optimal solution of the SDP relaxation happens to lie in a low dimensional space. Can an improved performance ratio be obtained for such instances? We show that the answer is yes in dimensions two and three and conjecture that this is also the case in any higher fixed dimension. In two dimensions an optimal 32/25+5√5-approximation algorithm was already obtained by Goemans. (Note that 32/25+5√5 ≃ 0.88456.) We obtain an alternative derivation of this result using Gegenbauer polynomials. Our main result is an improved rounding procedure for SDP solutions that lie in ℝ³ with a performance ratio of about 0.8818. The rounding procedure uses an interesting yin-yan coloring of the three dimensional sphere. The improved performance ratio obtained resolves, in the negative, an open problem posed by Feige and Schechtman [STOC'01]. They asked whether there are MAX CUT instances with integrality ratios arbitrarily close to α_GW ≃ 0.87856 that have optimal embedding, i.e., optimal solutions of their SDP relaxations, that lie in ℝ³.