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
Approximate graph coloring by semidefinite programming
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of 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)
Machine Learning
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
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
The RPR2 rounding technique for semidefinite programs
Journal of Algorithms
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
Quantum Information & Computation
The Grothendieck constant of random and pseudo-random graphs
Discrete Optimization
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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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 $$ {\rm Opt}_{p} (A) := \max \left\{ \sum_{i, j=1}^na_{ij}x_ix_j : (x_1,\ldots,x_n)\in {\mathbb{R}}^n \wedge \left(\sum_{i=1}^n |x_i|^p\right)^{1/p} \le 1 \right\}.$$ When p = 2 this is simply the problem of computing the maximum eigenvalue of A, whereas 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 Nemirovski et al. [Nemirovski, A., C. Roos, T. Terlaky. 1999. On maximization of quadratic form over intersection of ellipsoids with common center. Math. Program. Ser. A86(3) 463--473], Megretski [Megretski, A. 2001. Relaxations of quadratic programs in operator theory and system analysis. Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Vol. 129. Operator Theory Advances and Applications. Birkhäuser, Basel, 365--392], Charikar and Wirth [Charikar, M., A. Wirth. 2004. Maximizing quadratic programs: Extending Grothendieck's inequality. Proc. 45th Annual Sympos. Foundations Comput. Sci., IEEE Computer Society, 54--60] and was used in the work of Charikar and Wirth noted above, 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 × 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 Optp(A) up to a factor smaller than p/e + ¼. Hence as p → ∞ the UGC-hardness threshold for computing Optp(A) is exactly (p/e)(1 + o(1)).