Checking computations in polylogarithmic time
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Elements of information theory
Elements of information theory
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
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
Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
Some optimal inapproximability results
Journal of the ACM (JACM)
A new PCP outer verifier with applications to homogeneous linear equations and max-bisection
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Unique games on expanding constraint graphs are easy: extended abstract
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
Hardness of Solving Sparse Overdetermined Linear Systems: A 3-Query PCP over Integers
ACM Transactions on Computation Theory (TOCT)
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Linearity testing in characteristic two
IEEE Transactions on Information Theory - Part 1
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation resistance on satisfiable instances for predicates with few accepting inputs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be "non-trivial". Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1-δ fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a "margin" of Ω(√δ). We develop linearity and dictatorship testing procedures for functions f: Rn - R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.