SVP$_p$ is Deterministically NP-Hard for all $p > 2$, Even to Approximate Within a Factor of $2^{\log^{1-\varepsilon} n}$

TL;DR

This paper proves that the Shortest Vector Problem in $ ext{l}_p$ norms for $p > 2$ is NP-hard to approximate within a nearly exponential factor, establishing new hardness results for exact and approximate solutions.

Contribution

It provides the first proof of NP-hardness for exact SVP$_p$ in finite $ ext{l}_p$ norms and introduces elementary reduction techniques from PCP instances.

Findings

SVP$_p$ is NP-hard to approximate within $2^{ ext{log}^{1- ext{epsilon}} n}$ for all $p > 2$

Exact SVP$_p$ is NP-hard in finite $ ext{l}_p$ norms

Elementary reduction techniques using Vandermonde and Hadamard matrices

Abstract

We prove that SVP$_p$ is NP-hard to approximate within a factor of $2^{\log^{1 - \varepsilon} n}$, for all constants $\varepsilon > 0$ and $p > 2$, under standard deterministic Karp reductions. This result is also the first proof that \emph{exact} SVP$_p$ is NP-hard in a finite $\ell_p$ norm. Hardness for SVP$_p$ with $p$ finite was previously only known if NP $\not \subseteq$ 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, SVP$_p$ 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.