We show that the Closest Vector Problem with Preprocessing over `∞ norm (CVPP∞) is NP-hard to approximate to within a factor of (log n)1/2− , unless NP⊆ DTIME (2polylog(n)). The result is the same as that in [19] by Regev and Rosen, but our proof methods are different from theirs. Their reductions are based on norm embeddings. However, our reductions are based on the reduction of [2] and the property of Hadamard matrix.
We show that the Closest Vector Problem with Preprocessing over `∞ norm (CVPP∞) is NP-hard to approximate to within a factor of (log n)1/2− , unless NP⊆ DTIME (2polylog(n)). The result is the same as that in [19] by Regev and Rosen, but our proof methods are different from theirs. Their reductions are based on norm embeddings. However, our reductions are based on the reduction of [2] and the property of Hadamard matrix.
We prove two deterministic inapproximability results. First, for every fixed 0<ϵ<1/2, Euclidean GapCVP(2) is NP-hard with gap factor n1/2−ϵ under deterministic polynomial-time many-one reductions, where n denotes the lattice rank. Consequently, the Euclidean closest vector problem is NP-hard to approximate within the same factor. This improves the previous n1/400 hardness factor in Chapter 7 of the OpenAI report [Ope25]. Aharonov and Regev gave short certificates for both the YES and the NO case of GapCVP(2) at gap factor Cn, placing that problem in NP∩coNP for an absolute constant C>0 [AR05]. An NP-hard problem lying in coNP would give NP=coNP, so the factor n1/2−ϵ above cannot be improved to Cn unless the two classes coincide. Second, for every fixed 0<ϵ<1, the gap versions of binary nearest codeword and binary syndrome decoding are NP-hard with factor n1−ϵ under deterministic polynomial-time many-one reductions, where n denotes the binary block length. Consequently, both optimization problems are NP-hard to approximate within the same factor. This improves the previous n1/200 hardness factor in Chapter 7 of the OpenAI report [Ope26].
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.>
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 .
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.
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.