# The hardness of approximate optima in lattices, codes, and systems of linear equations

@article{Arora1993TheHO, title={The hardness of approximate optima in lattices, codes, and systems of linear equations}, author={Sanjeev Arora and L. Babai and J. Stern and E. Sweedyk}, journal={Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science}, year={1993}, pages={724-733} }

We prove the following about the Nearest Lattice Vector Problem (in any l/sub p/ norm), the Nearest Code-word Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some /spl epsiv/>0 there exists a polynomial time algorithm that approximates the optimum within a factor of 2/sup log(0.5-/spl epsiv/)/ /sup n/ then NP is in quasi-polynomial deterministic time… Expand

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