Closed form two-sided bounds for probabilities that at least r and exactly r out of n events occur
Mathematics of Operations Research
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Some optimal inapproximability results
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
Improved Rounding Techniques for the MAX 2-SAT and MAX DI-CUT Problems
Proceedings of the 9th International IPCO Conference on Integer Programming and Combinatorial Optimization
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
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
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Approximation Resistant Predicates from Pairwise Independence
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
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The results of Raghavendra [2008] show that assuming Khot’s Unique Games Conjecture [2002], for every constraint satisfaction problem there exists a generic semidefinite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. Khot et al. [2004] established a general approach for determining the exact approximation factor and the corresponding optimal rounding algorithm for any given constraint satisfaction problem. However, this approach crucially relies on results explicitly proving optimal partitions in the Gaussian space. Until recently, Borell’s result [1985] was the only nontrivial Gaussian partition result known. In this article we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel [2012]. This Gaussian result allows us to determine the exact Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick’s algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is ≈ 0.796 as conjectured by Zwick [1998]. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP: Using the well-known fact that jointly normal pairwise independent random variables are fully independent, we show that the UGC hardness of Max-k-CSP is ⌈(k+1)/2⌉ 2k−1, improving on results of Austrin and Mossel [2009].