Mathematical Programming: Series A and B
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)
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
How Good is the Goemans--Williamson MAX CUT Algorithm?
SIAM Journal on Computing
Improved approximation of max-cut on graphs of bounded degree
Journal of Algorithms
Derandomizing semidefinite programming based approximation algorithms
FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
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
Geometry of Cuts and Metrics
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Noise stability of functions with low in.uences invariance and optimality
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The RPR2 rounding technique for semidefinite programs
Journal of Algorithms
Spherical basis functions and uniform distribution of points on spheres
Journal of Approximation Theory
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
Graph expansion and the unique games conjecture
Proceedings of the forty-second ACM symposium on Theory of computing
Towards Sharp Inapproximability for Any 2-CSP
SIAM Journal on Computing
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Subsampling mathematical relaxations and average-case complexity
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Small bipartite subgraph polytopes
Operations Research Letters
A hybridization between memetic algorithm and semidefinite relaxation for the max-cut problem
Proceedings of the 14th annual conference on Genetic and evolutionary computation
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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 α . sdp, where α ∼ 0.87856 and sdp is the value of the semidefinite program, which is an upper bound on opt, the number of edges in the maximum cut. This manuscript shows the following results: (1) The integrality ratio of the semidefinite program is α. The previously known bound on the integrality 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. (3) There are graphs and optimal embeddings for which the best hyperplane approximates opt within a ratio no better than α, even in the presence of additional valid constraints. This strengthens a result of Karloff that applied only to the expected number of edges cut by a random hyperplane.