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
Journal of Algorithms
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
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Semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio techniques. Are the bounds obtained in this way tight? This problem was considered before by Karloff [SIAM J. Comput., 29 (1999), pp. 336--350] and by Alon and Sudakov [ Combin. Probab. Comput., 9 (2000), pp. 1--12]. Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut in the sense that the expected value of the solutions obtained by the algorithms may be as small as the analyses ensure. We also obtain similar results for a related problem. Our approach is quite general and could possibly be applied to some additional problems and algorithms.