The hardness of approximation: gap location
Computational Complexity
Some optimal inapproximability results
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Improved low-degree testing and its applications
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
A threshold of ln n for approximating set cover
Journal of the ACM (JACM)
On the approximability of the traveling salesman problem (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Relations between average case complexity and approximation complexity
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Robust Characterizations of Polynomials withApplications to Program Testing
SIAM Journal on Computing
Hardness Results for Coloring 3 -Colorable 3 -Uniform Hypergraphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
Hardness of Approximating the Minimum Distance of a Linear Code
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A polylogarithmic approximation of the minimum bisection
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Gadgets Approximation, and Linear Programming
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Clique is hard to approximate within n1-
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
On hardness of learning intersection of two halfspaces
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
The structure of winning strategies in parallel repetition games
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
On the hardness of learning intersections of two halfspaces
Journal of Computer and System Sciences
NP-hardness of approximately solving linear equations over reals
Proceedings of the forty-third annual ACM symposium on Theory of computing
On unique games with negativeweights
COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
Approximating CSPs with global cardinality constraints using SDP hierarchies
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On the complexity of global constraint satisfaction
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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We show an optimal hardness result for the following problem: Given a system of homogeneous linear equations over GF(2) with 3 variables per equation, find a balanced assignment that satisfies maximum number of equations. For arbitrarily small constant ζ 0, we show that it is hard to determine (in polynomial time) whether such a system has a balanced assignment that satisfies 1-ζ fraction of equations or there is no balanced assignment that satisfies more than ½+ζ fraction of equations. As a corollary, we show that it is hard to approximate (in polynomial time) the Max-Bisection problem within factor 16⁄15-ζ. These hardness results hold under the assumption NP ⊈ ∩ε 0 DTIME(2nε).Our results are obtained via a construction of a new PCP outer verifier that has a mixing property and a smoothness property. These properties are crucial in the analysis of the inner verifier. No previous outer verifier can achieve both these properties simultaneously. An outer verifier is essentially a 2-query PCP over a large alphabet. Loosely speaking, the mixing property says that the locations of the two queries read by the verifier are uncorrelated. The smoothness property says that the verifier's acceptance predicate is close to being a bijective predicate. Our construction relies on the algebraic techniques used to prove the PCP Theorem. This is in contrast with all earlier constructions that use the PCP Theorem as a black-box. The progress in inapproximability theory seems to require new ideas for building outer verifiers and our construction takes a first step in that direction.