Approximating the SVP to within a Factor (1+1/dimε) Is NP-Hard under Randomized Reductions

摘要

Recently Ajtai showed that to approximate the shortest lattice vector in the l2-norm within a factor (1+2−dimk), for a sufficiently large constant k, is NP-hard under randomized reductions. We improve this result to show that to approximate a shortest lattice vector within a factor (1+dim−ε), for any ε>0, is NP-hard under randomized reductions. Our proof also works for arbitrary lp-norms, 1⩽p<∞.