Balanced max 2-sat might not be the hardest
Proceedings of the thirty-ninth 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
Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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
3-bit dictator testing: 1 vs. 5/8
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Towards computing the Grothendieck constant
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
The UGC Hardness Threshold of the Lp Grothendieck Problem
Mathematics of Operations Research
Towards Sharp Inapproximability for Any 2-CSP
SIAM Journal on Computing
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
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Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson [GW95] semidefinite programming (SDP) algorithm is guaranteed to find a cut of size .878 · c. However this guarantee becomes trivial when c is near 1/2, since a random cut has expected size 1/2. Recently, Charikar and Worth [CW04] (analyzing an algorithm of Feige and Langberg [FL01]) showed that given a graph with maximum cut 1/2 + \in, one can find a cut of size 1/2 + \Omega(\in/ log(1/\in)). The main contribution of our paper is twofold:1. We give a natural 1/2+\in vs. 1/2+O(\in/ log(1/\in)) SDP gap for MAXCUT in Gaussian space. This shows that the SDP-rounding algorithm of Charikar-Worth is essentially best possible. Further, the "s-linear rounding functions" used in [CW04, FL01] arise as optimizers in our analysis, somewhat confirming a suggestion of [FL01]. 2. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the Charikar-Worth guarantee with any efficient algorithm is NP-hard, assuming the Unique Games Conjecture (UGC) [Kho02]. We view this result as essentially settling the approximability of MAXCUT, assuming UGC. Building on (1) we show how "randomness reduction" on related SDP gaps for the QUADRATICPROGRAMMING programming problem lets us make the \Omega(log(1/\in)) gap as large as \Omega(log n) for n-vertex graphs. In addition to optimally answering an open question of [AMMN06], this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our technique in (2), we also show how to translate Reeds's [Ree93] SDP gap for the Grothendieck Inequality into a UGChardness result for computing the \parallel · \parallel\infty \mapsto 1 norm of a matrix.