Two-prover one-round proof systems: their power and their problems (extended abstract)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Combinatorial properties and complexity of a max-cut approximation
European Journal of Combinatorics
Laplacian eigenvalues and the maximum cut problem
Mathematical Programming: Series A and B
Free Bits, PCPs, and Nonapproximability---Towards Tight Results
SIAM Journal on Computing
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
P-Complete Approximation Problems
Journal of the ACM (JACM)
How Good is the Goemans--Williamson MAX CUT Algorithm?
SIAM Journal on Computing
Gadgets, Approximation, and Linear Programming
SIAM Journal on Computing
Some optimal inapproximability results
Journal of the ACM (JACM)
Constructing Worst Case Instances for Semidefinite Programming Based Approximation Algorithms
SIAM Journal on Discrete Mathematics
Improved approximation of max-cut on graphs of bounded degree
Journal of Algorithms
On the optimality of the random hyperplane rounding technique for max cut
Random Structures & Algorithms - Probabilistic methods in combinatorial optimization
Bipartite Subgraphs and the Smallest Eigenvalue
Combinatorics, Probability and Computing
Machine Learning
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
On the advantage over a random assignment
Random Structures & Algorithms
Journal of Algorithms
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
On Non-Approximability for Quadratic Programs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Approximating the Cut-Norm via Grothendieck's Inequality
SIAM Journal on Computing
The RPR2 rounding technique for semidefinite programs
Journal of Algorithms
SDP gaps and UGC-hardness for MAXCUTGAIN
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Balanced max 2-sat might not be the hardest
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Towards Sharp Inapproximability For Any 2-CSP
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
A Branch and Bound Algorithm for Max-Cut Based on Combining Semidefinite and Polyhedral Relaxations
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
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
3-bit dictator testing: 1 vs. 5/8
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Max cut and the smallest eigenvalue
Proceedings of the forty-first annual ACM symposium on Theory of computing
Conditional hardness for satisfiable 3-CSPs
Proceedings of the forty-first annual ACM symposium on Theory of computing
Approximating linear threshold predicates
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
Towards Sharp Inapproximability for Any 2-CSP
SIAM Journal on Computing
Approximating Linear Threshold Predicates
ACM Transactions on Computation Theory (TOCT)
A new point of NP-hardness for unique games
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
A hybridization between memetic algorithm and semidefinite relaxation for the max-cut problem
Proceedings of the 14th annual conference on Genetic and evolutionary computation
On the usefulness of predicates
ACM Transactions on Computation Theory (TOCT)
Majority is stablest: discrete and SoS
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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Let G be an undirected graph for which the standard Max-Cut SDP relaxation achieves at least a c fraction of the total edge weight, 1/2 [1/2,1] by GapSDP(c) = inf{s : (c, s) is an SDP gap}. In this paper we complete a long line of work [15, 14, 20, 36, 19, 17, 13, 28] by determining the entire SDP gap curve; we show GapSDP(c) = S(c) for a certain explicit (but complicated to state) function S. In particular, our lower bound GapSDP(c) - S(c) is proved via a polynomial-time - RPR2' algorithm. Thus we have given an efficient, optimal SDP-rounding algorithm for Max-Cut. The fact that it is RPR2 confirms a conjecture of Feige and Langberg [17]. We also describe and analyze the tight connection between SDP gaps and Long Code tests (and the constructions of [25, 3, 4]). Using this connection, we give optimal Long Code tests for Max-Cut. Combining these with results implicit in [27, 29] and ideas from [19], we derive the following conclusions: - The Max-Cut SDP gap curve subject to triangle inequalities is also given by S(c). - No RPR2 algorithm can be guaranteed to find cuts of value larger than S(c) in graphs where the optimal cut is c. (Contrast this with the fact that in the graphs exhibiting the c vs. S(c) SDP gap, our RPR2 algorithm actually finds the optimal cut.) - Further, no polynomial-time algorithm of any kind can have such a guarantee, assuming P ≠ NP and the Unique Games Conjecture.