Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Property testing and its connection to learning and approximation
Journal of the ACM (JACM)
How Good is the Goemans--Williamson MAX CUT Algorithm?
SIAM Journal on Computing
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)
Bipartite Subgraphs and the Smallest Eigenvalue
Combinatorics, Probability and Computing
On semidefinite programming relaxations for graph coloring and vertex cover
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved approximation of max-cut on graphs of bounded degree
Journal of Algorithms
Random Structures & Algorithms - Probabilistic methods in combinatorial optimization
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Linear programming relaxations of maxcut
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Rounding two and three dimensional solutions of the SDP relaxation of MAX CUT
APPROX'05/RANDOM'05 Proceedings of the 8th international workshop on Approximation, Randomization and Combinatorial Optimization Problems, and Proceedings of the 9th international conference on Randamization and Computation: algorithms and techniques
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MAX CUT is the problem of partitioning the vertices of a graph into two sets, maximizing the number of edges joining these sets. This problem is NP-hard. Goemans and Williamson proposed an algorithm that first uses a semidefinite programming relaxation of MAX CUT to embed the vertices of the graph on the surface of an n dimensional sphere, and then uses a random hyperplane to cut the sphere in two, giving a cut of the graph. They show that the expected number of edges in the random cut is at least &agr; \cdot sdp, where &agr; \simeq 0.87856 and sdp is the value of the semidefinite program.This manuscript shows the following results:1. The integrality ratio of the semidefinite program is &agr;. The previously known bound on theintegrality ratio was roughly 0.8845.2. In the presence of the so called “triangle constraints”, the integrality ratio is no better than roughly 0.891. The previously known bound was above 0.95.