Self-testing/correcting with applications to numerical problems
Journal of Computer and System Sciences - Special issue: papers from the 22nd ACM symposium on the theory of computing, May 14–16, 1990
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
Free Bits, PCPs, and Nonapproximability---Towards Tight Results
SIAM Journal on Computing
On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems
Theoretical Computer Science
Some optimal inapproximability results
Journal of the ACM (JACM)
Randomness-efficient low degree tests and short PCPs via epsilon-biased sets
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Guest column: inapproximability results via Long Code based PCPs
ACM SIGACT News
On the hardness of approximating Max-Satisfy
Information Processing Letters
Hardness of Learning Halfspaces with Noise
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
New Results for Learning Noisy Parities and Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
NP-hardness of approximately solving linear equations over reals
Proceedings of the forty-third annual ACM symposium on Theory of computing
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A classic result due to Håstad established that for every constant ϵ 0, given an overdetermined system of linear equations over a finite field Fq where each equation depends on exactly 3 variables and at least a fraction (1 − ϵ) of the equations can be satisfied, it is NP-hard to satisfy even a fraction (1/q + ϵ) of the equations. In this work, we prove the analog of Håstad’s result for equations over the integers (as well as the reals). Formally, we prove that for every ϵ, δ 0, given a system of linear equations with integer coefficients where each equation is on 3 variables, it is NP-hard to distinguish between the following two cases: (i) there is an assignment of integer values to the variables that satisfies at least a fraction (1 − ϵ) of the equations, and (ii) no assignment even of real values to the variables satisfies more than a fraction δ of the equations.