Hardness of the Binary Covering Radius Problem in Large $\ell_p$ Norms

TL;DR

This paper proves the NP-hardness of approximating the Binary Covering Radius Problem in large $p$ norms, establishing explicit hardness thresholds for a range of $p$ values and extending previous results to new variants.

Contribution

It establishes the first explicit NP-hardness results for $p$-norm GapCRP for certain p, and connects these results to the Binary Covering Radius Problem and Linear Discrepancy.

Findings

NP-hardness of $p$-norm GapCRP for p > 35.31

Explicit hardness threshold $rac{9}{8}$ for large p

Hardness results for the Binary Covering Radius Problem and Linear Discrepancy

Abstract

We study the hardness of the $\gamma$-approximate decisional Covering Radius Problem on lattices in the $\ell_p$ norm ($\gamma$-$\text{GapCRP}_p$). Specifically, we prove that there is an explicit function $\gamma(p)$, with $\gamma(p) > 1$ for $p > p_0 \approx 35.31$ and $\lim_{p \to \infty} \gamma(p) = 9/8$, such that for any constant $\varepsilon > 0$, $(\gamma(p) - \varepsilon)$-$\text{GapCRP}_p$ is $\mathsf{NP}$-hard. This shows the first hardness of $\text{GapCRP}_p$ for explicit $p < \infty$. Work of Haviv and Regev (CCC, 2006 and CJTCS, 2012) previously showed $\Pi_2$-hardness of approximation for $\text{GapCRP}_p$ for all sufficiently large (but non-explicit) finite $p$ and for $p = \infty$. In fact, our hardness results hold for a variant of $\text{GapCRP}$ called the Binary Covering Radius Problem ($\text{BinGapCRP}$), which trivially reduces to both $\text{GapCRP}$ and the decisional Linear Discrepancy Problem ($\text{LinDisc}$) in any norm in an approximation-preserving way. We also show $\Pi_2$-hardness of $(9/8 - \varepsilon)$-$\text{BinGapCRP}$ in the $\ell_{\infty}$ norm for any constant $\varepsilon > 0$. Our work extends and heavily uses the work of Manurangsi (IPL, 2021), which showed $\Pi_2$-hardness of $(9/8 - \varepsilon)$-$\text{LinDisc}$ in the $\ell_{\infty}$ norm.