The Hardness of Approximate Optimia in Lattices, Codes, and Systems of Linear Equations
نویسندگان
چکیده
We prove the following about the Nearest Lattice Vector Problem (an any e, norm), the Nearest Codeword 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 6 > 0 there exists a polynomial t ime algorithm that approximates the optimum within a factor of 210g0'5-'n then N P is in quasi-polynomial deterministic t ime: N P DTIME( npoiy(logn)). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the em norm. Improving the factor 2log0"-' " to JdTm for either of the lattice problems would imply the hardness of the Shortest Vector Problem in e 2 norm; an old open problem. Our proofs use reductions from few-prover, oneround interactive proof systems [FL], [BG+], either directly, or through a set-cover problem.
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