Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
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
A 7/8-Approximation Algorithm for MAX 3SAT?
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Randomly supported independence and resistance
Proceedings of the forty-first annual ACM symposium on Theory of computing
Optimal Sherali-Adams Gaps from Pairwise Independence
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP
STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
Approximating Linear Threshold Predicates
ACM Transactions on Computation Theory (TOCT)
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A predicate P:{– 1, 1}k→{0,1} can be associated with a constraint satisfaction problem $\textsc{Max CSP}{(P)}$. P is called “approximation resistant” if $\textsc{Max CSP}{(P)}$ cannot be approximated better than the approximation obtained by choosing a random assignment, and “approximable” otherwise. This classification of predicates has proved to be an important and challenging open problem. Motivated by a recent result of Austrin and Mossel (Computational Complexity, 2009), we consider a natural subclass of predicates defined by signs of quadratic polynomials, including the special case of predicates defined by signs of linear forms, and supply algorithms to approximate them as follows. In the quadratic case we prove that every symmetric predicate is approximable. We introduce a new rounding algorithm for the standard semidefinite programming relaxation of $\textsc{Max CSP}{(P)}$ for any predicate P:{– 1, 1}k→{0,1} and analyze its approximation ratio. Our rounding scheme operates by first manipulating the optimal SDP solution so that all the vectors are nearly perpendicular and then applying a form of hyperplane rounding to obtain an integral solution. The advantage of this method is that we are able to analyze the behaviour of a set of k rounded variables together as opposed to just a pair of rounded variables in most previous methods. In the linear case we prove that a predicate called “Monarchy” is approximable. This predicate is not amenable to our algorithm for the quadratic case, nor to other LP/SDP-based approaches we are aware of.