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Annual Symposium on Foundations of Computer Science

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

Sanjeev Arora, László Babai, Jacques Stern, Z. Sweedyk

Published Dec 30, 2002• 181 citations

Abstract

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: NP/spl sube/DTIME(n/sup poly(log/ /sup n)/). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l/sub /spl infin// norm. Improving the factor 2/sup log(0.5-/spl epsiv/)/ /sup n/ to /spl radic/(dim) for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l/sub 2/ 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.>

DOI: 10.1109/sfcs.1993.366815

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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: NP/spl sube/DTIME(n/sup poly(log/ /sup n)/). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the l/sub /spl infin// norm. Improving the factor 2/sup log(0.5-/spl epsiv/)/ /sup n/ to /spl radic/(dim) for either of the lattice problems would imply the hardness of the Shortest Vector Problem in l/sub 2/ 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.>

We show that the minimum distance of a linear code (or equivalently, the weight of the lightest codeword) is not approximable to within any constant factor in random polynomial time (RP), unless NP equals RP. Under the stronger assumption that NP is not contained in RQP (random quasi-polynomial time), we show that the minimum distance is not approximable to within the factor 2/sup log(1-/spl epsiv/)n/, for any /spl epsiv/>0, where n denotes the block length of the code. Our results hold for codes over every finite field, including the special case of binary codes. In the process we show that the nearest codeword problem is hard to solve even under the promise that the number of errors is (a constant factor) smaller than the distance of the code. This is a particularly meaningful version of the nearest codeword problem. Our results strengthen (though using stronger assumptions) a previous result of A. Vardy (1997) who showed that the minimum distance is NP-hard to compute exactly. Our results are obtained by adapting proofs of analogous results for integer lattices due to M. Ajtai (1998) and D. Micciancio (1998). A critical component in the adaptation is our use of linear codes that perform better than random (linear) codes.

Abstract: We prove that the Shortest Vector Problem (SVP) on point lattices is NPhard to approximate for any constant factor under polynomial-time randomized reductions with one-sided error that produce no false positives. We also prove inapproximability for quasi-polynomial factors under the same kind of reductions running in subexponential time. Previous hardness results for SVP either incurred two-sided error, or only proved hardness for small constant approximation factors. Close similarities between our reduction and recent results on the complexity of the analogous problem for linear codes make our new proof an attractive target for derandomization, paving the road to a possible NP-hardness proof for SVP under deterministic polynomial-time reductions.

Let p > 1 be any fixed real. We show that assuming NP ⊈ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in ℓ p norm within a constant factor. Under the stronger assumption NP ⊈ RTIME(2 poly (log n ) ), we show that there is no polynomial-time algorithm with approximation ratio 2 (log n ) 1/2−ϵ where n is the dimension of the lattice and ϵ > 0 is an arbitrarily small constant.We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product , a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2 (log n ) 1/2-ϵ .

We show simple constant-round interactive proof systems for problems capturing the approximability, to within a factor of n , of optimization problems in integer lattices, specifically, the closest vector problem (CVP) and the shortest vector problem (SVP). These interactive proofs are for the coNP direction; that is, we give an interactive protocol showing that a vector is far from the lattice (for CVP) and an interactive protocol showing that the shortest-lattice-vector is long (for SVP). Furthermore, these interactive proof systems are honest-verifier perfect zero-knowledge. We conclude that approximating CVP (resp., SVP) within a factor of n is in N P ∩co A M . Thus, it seems unlikely that approximating these problems to within a n factor is NP-hard. Previously, for the CVP (resp., SVP) problem, Lagarias et al. (1990, Combinatorica 10 , 333–348), Håstad (1988, Combinatorica 8 , 75–81), and Banaszczyk (1993, Math. Annal. 296 , 625–635) showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in N P ∩co N P . On the other hand, Arora et al. (1997, J. Comput. System Sci. 54 , 317–331) showed that the gap problem corresponding to approximating CVP within 2 log 0.999 n is quasi-NP-hard.