Combinatorial properties and complexity of a max-cut approximation
European Journal of Combinatorics
Laplacian eigenvalues and the maximum cut problem
Mathematical Programming: Series A and B
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
On the integrality ratio of semidefinite relaxations of MAX CUT
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
Computer assisted proof of optimal approximability results
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
The RPR2 Rounding Technique for Semidefinite Programs
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
63-Approximation Algorithm for MAX DICUT
APPROX '01/RANDOM '01 Proceedings of the 4th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and 5th International Workshop on Randomization and Approximation Techniques in Computer Science: Approximation, Randomization and Combinatorial Optimization
Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT
ISTCS '95 Proceedings of the 3rd Israel Symposium on the Theory of Computing Systems (ISTCS'95)
Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Optimal Inapproximability Results for Max-Cut and Other 2-Variable CSPs?
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
An optimal sdp algorithm for max-cut, and equally optimal long code tests
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
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Goemans and Williamson obtained an approximation algorithm for the MAX CUT problem with a performance ratio of αGW≃ 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 $\frac{32}{25+5\sqrt{5}}$-approximation algorithm was already obtained by Goemans. (Note that $\frac{32}{25+5\sqrt{5}} \simeq 0.88456$.) 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 αGW≃ 0.87856 that have optimal embedding, i.e., optimal solutions of their SDP relaxations, that lie in ℝ3.