Constructing worst case instances for semidefinite programming based approximation algorithms

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.