We show that unless NP ⊆ RTIME (2poly(log n)), for any ε > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe lp norm (1 ≤q p<∞) to within a factor of 2(log n)1-ε. This improves the previous best factor of 2(logn)1/2-ε under the same complexity assumption due to Khot. Under the stronger assumption NP ࣰ RSUBEXP, we obtain a hardness factor of nc/log log n for some c > 0.
We show that unless NP ⊆ RTIME (2poly(log n)), for any ε > 0 there is no polynomial-time algorithm approximating the Shortest Vector Problem (SVP) on n-dimensional lattices inthe lp norm (1 ≤q p<∞) to within a factor of 2(log n)1-ε. This improves the previous best factor of 2(logn)1/2-ε under the same complexity assumption due to Khot. Under the stronger assumption NP ࣰ RSUBEXP, we obtain a hardness factor of nc/log log n for some c > 0.
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 .
$ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RP}{\mathsf{RP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\poly}{\mathop{\mathrm{poly}}} \newcommand{\eps}{\varepsilon} $ We show that unless $\NP \subseteq \RTIME (2^{\poly(\log{n})})$, there is no polynomial-time algorithm approximating the Shortest Vector Problem ($\SVP$) on n-dimensional lattices in the ℓp norm (1≤p<∞) to within a factor of $2^{(\log{n})^{1-\eps}}$ for any $\eps >0$. This improves the previous best factor of $2^{(\log{n})^{1/2-\eps}}$ under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption $\NP \nsubseteq \RSUBEXP$, we obtain a hardness factor of nc/loglogn for some c>0. Our proof starts with Khot's $\SVP$ instances that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novel part is in the analysis, where we show that the lattices of Khot behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski.
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.
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 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.
SVPp is Deterministically NP-Hard for all p>2, Even to Approximate Within a Factor of 2log1−εn
We prove that SVPp is NP-hard to approximate within a factor of 2log1−εn, for all constants ε>0 and p>2, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVPp is NP-hard in a finite ℓp norm. Hardness for SVPp with p finite was previously only known if NP ⊆ RP, and under that assumption, hardness of approximation was only known for all constant factors. As a corollary to our main theorem, we show that under the Sliding Scale Conjecture, SVPp is NP-hard to approximate within a small polynomial factor, for all constants p>2. Our proof techniques are surprisingly elementary; we reduce from a \emph{regularized} PCP instance directly to the shortest vector problem by using simple gadgets related to Vandermonde matrices and Hadamard matrices.