Constructing worst case instances for semidefinite programming based approximation algorithms

Authors:
Noga Alon;Benny Sudakov;Uri Zwick
Affiliations:
Department of Mathematics, Raymond and Beverly Sackler, Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Mathematics, Princeton University, Princeton, NJ and Institute for Advanced Study, Princeton, NJ;Department of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Venue:
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Year:
2001

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Abstract

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 and by Alon and Sudakov. 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. We also obtain similar results for several related problems. Our approach is quite general and could possibly be applied to some additional problems and algorithms.