Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
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Approximating the cut-norm via Grothendieck's inequality
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Machine Learning
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Optimal Inapproximability Results for Max-Cut and Other 2-Variable CSPs?
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
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
Correlation clustering with a fixed number of clusters
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
On the fourier tails of bounded functions over the discrete cube
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The UGC hardness threshold of the ℓp Grothendieck problem
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
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
Semidefinite Programming Heuristics for Surface Reconstruction Ambiguities
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part I
Towards computing the Grothendieck constant
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
The UGC Hardness Threshold of the Lp Grothendieck Problem
Mathematics of Operations Research
Simulating Quantum Correlations with Finite Communication
SIAM Journal on Computing
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Approximating integer quadratic programs and MAXCUT in subdense graphs
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Algorithms and hardness for subspace approximation
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On quadratic programming with a ratio objective
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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This paper studies the computational complexity of the following type of quadratic programs: given an arbitrary matrix whose diagonal elements are zero, find \chi \varepsilon {-1, 1}^n that maximizes \chi ^{\rm T} Mx this problem recently attracted attention due to its application in various clustering settings, as well as an intriguing connection to the famous Grothendieck inequality. It is approximable to within a factor of O(log n), and known to be NP-hard to approximate within any factor better than 13/110 - \varepsilon for all \varepsilon O. We show showthat it is quasi-NP-hard to approximate to a factor better than O(Log^\gamman) for some \gamma 0.The integrality gap of the natural semide?nite relaxation for this problem is known as the Grothendieck constant the complete graph, and known to be \theta (\log n). The proof this fact was nonconstructive, and did not yield an explicit problem instance where this integrality gap is achieved. Our techniques yield an explicit instance for which the integrality gap is \Omega (\frac{{\log n}}{{\log \log n}}), essentially answering one of the open problems of Alon et al. [AMMN].