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
The complexity of approximating a nonlinear program
Mathematical Programming: Series A and B
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th 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
Near-optimal algorithms for unique games
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A PTAS for the minimization of polynomials of fixed degree over the simplex
Theoretical Computer Science - Approximation and online algorithms
SDP gaps and UGC-hardness for MAXCUTGAIN
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The Grothendieck constant of random and pseudo-random graphs
Discrete Optimization
Towards computing the Grothendieck constant
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Algorithms and hardness for subspace approximation
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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Hypercontractivity, sum-of-squares proofs, and their applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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For p ≥ 2 we consider the problem of, given an n × n matrix A = (aij) whose diagonal entries vanish, approximating in polynomial time the number {display equation} (where optimization is taken over real numbers). When p = 2 this is simply the problem of computing the maximum eigenvalue of A, while for p = ∞ (actually it suffices to take p ≈ log n) it is the Grothendieck problem on the complete graph, which was shown to have a O(log n) approximation algorithm in [27, 26, 15], and was used in [15] to design the best known algorithm for the problem of computing the maximum correlation in Correlation Clustering. Thus the problem of approximating Optp (A) interpolates between the spectral (p = 2) case and the Correlation Clustering (p = ∞) case. From a physics point of view this problem corresponds to computing the ground states of spin glasses in a hard-wall potential well. We design a polynomial time algorithm which, given p ≥ 2 and an n x n matrix A = (aij) with zeros on the diagonal, computes Optp (A) up to a factor p/e + 30 log p. On the other hand, assuming the unique games conjecture (UGC) we show that it is NP-hard to approximate (1.2) up to a factor smaller than p/e + 1/4. Hence as p → ∞ the UGC-hardness threshold for computing Optp (A) is exactly p/e (1 + o(1)).