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

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4 Author(s)
S. Arora ; Div. of Comput. Sci., California Univ., Berkeley, CA, USA ; L. Babai ; J. Stern ; Z. Sweedyk

We prove the following about the Nearest Lattice Vector Problem (in any lp 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 ε>0 there exists a polynomial time algorithm that approximates the optimum within a factor of 2log(0.5-ε) n then NP is in quasi-polynomial deterministic time: NP⊆DTIME(npoly(log n)). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l norm. Improving the factor 2log(0.5-ε) n to √(dim) for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l2 norm; an old open problem. Our proofs use reductions from few-prover, one-round interactive proof systems, either directly, or through a set-cover problem

Published in:

Foundations of Computer Science, 1993. Proceedings., 34th Annual Symposium on

Date of Conference:

3-5 Nov 1993