We study the hardness of the $γ$-approximate decisional Covering Radius Problem on lattices in the $\ell_p$ norm ($γ$-$\text{GapCRP}_p$). Specifically, we prove that there is an explicit function $γ(p)$, with $γ(p) > 1$ for $p > p_0 \approx 35.31$ and $\lim_{p \to \infty} γ(p) = 9/8$, such that for any constant $\varepsilon > 0$, $(γ(p) - \varepsilon)$-$\text{GapCRP}_p$ 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 $Π_2$-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 $Π_2$-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 $Π_2$-hardness of $(9/8 - \varepsilon)$-$\text{LinDisc}$ in the $\ell_{\infty}$ norm.

翻译：我们研究格上 $\ell_p$ 范数中 $\gamma$-近似判定覆盖半径问题 ($\gamma$-$\text{GapCRP}_p$) 的硬度。具体而言，我们证明存在一个显式函数 $\gamma(p)$，满足当 $p > p_0 \approx 35.31$ 时 $\gamma(p) > 1$ 且 $\lim_{p \to \infty} \gamma(p) = 9/8$，使得对任意常数 $\varepsilon > 0$，$(γ(p) - \varepsilon)$-$\text{GapCRP}_p$ 是 $\mathsf{NP}$-难的。这一结果首次展示了对于显式有限 $p$ 值下 $\text{GapCRP}_p$ 的硬度。此前 Haviv 与 Regev 的工作（CCC, 2006 和 CJTCS, 2012）仅证明了对于所有充分大（但非显式）的有限 $p$ 以及 $p = \infty$ 情形下 $\text{GapCRP}_p$ 近似的 $Π_2$-难度。实际上，我们的硬度结果适用于 $\text{GapCRP}$ 的一个变体，称为二元覆盖半径问题 ($\text{BinGapCRP}$)，该问题可在任意范数下以保持近似的方式平凡归约到 $\text{GapCRP}$ 和判定线性偏差问题 ($\text{LinDisc}$)。我们还证明在 $\ell_{\infty}$ 范数中，对任意常数 $\varepsilon > 0$，$(9/8 - \varepsilon)$-$\text{BinGapCRP}$ 具有 $Π_2$-难度。我们的工作扩展并深入利用了 Manurangsi 的工作（IPL, 2021），该工作证明了在 $\ell_{\infty}$ 范数中 $(9/8 - \varepsilon)$-$\text{LinDisc}$ 的 $Π_2$-难度。