TY - JOUR

T1 - The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations

AU - Arora, Sanjeev

AU - Babai, László

AU - Stern, Jacques

AU - Sweedyk, Z.

N1 - Funding Information:
* Research done while the author was at UC Berkeley, and was supported by an IBM Graduate Fellowship. -Partially supported by NSF grant CCR-9014562. Partially supported by NSF grant CCR-9201092 and Sandia National Laboratories.

PY - 1997/4

Y1 - 1997/4

N2 - We prove the following about the Nearest Lattice Vector Problem (in any lp 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 A/P-hard. 2. If for some ε > 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2log0.5-ε n, then every NP language can be decided in quasi-polynomial deterministic time, i.e., NP ⊆ DTIME(npoly(log n)). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l∞ norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2log1-ε n or nε. Improving the factor 2log0.5-ε n to √dimension 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 [FL], BG+], either directly, or through a set-cover problem.

AB - We prove the following about the Nearest Lattice Vector Problem (in any lp 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 A/P-hard. 2. If for some ε > 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2log0.5-ε n, then every NP language can be decided in quasi-polynomial deterministic time, i.e., NP ⊆ DTIME(npoly(log n)). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l∞ norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2log1-ε n or nε. Improving the factor 2log0.5-ε n to √dimension 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 [FL], BG+], either directly, or through a set-cover problem.

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U2 - 10.1006/jcss.1997.1472

DO - 10.1006/jcss.1997.1472

M3 - Article

AN - SCOPUS:0031119485

SN - 0022-0000

VL - 54

SP - 317

EP - 331

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

IS - 2

ER -