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

2026-03-03

AI Summary

This work investigates the computational and approximation complexity of the Binary Gap Covering Radius Problem (BinGapCRP) on lattices under large $\ell_p$ norms. By constructing an explicit function $\gamma(p)$, it establishes the first NP-hardness result for $(\gamma(p) - \varepsilon)$-BinGapCRP in the $\ell_p$ norm when $p > p_0 \approx 35.31$, and proves $\Pi_2$-hardness in the $\ell_\infty$ norm. This yields the first explicit NP-hardness for GapCRP at any finite $p < \infty$ and extends the known $\Pi_2$-hardness of the Linear Discrepancy (LinDisc) problem to BinGapCRP. Technically, the proof combines Manurangsi’s $\Pi_2$-hardness framework with approximation-preserving reductions, establishing a lower bound on the approximability of BinGapCRP that asymptotically approaches $9/8$, thereby strengthening its theoretical connections to both LinDisc and GapCRP.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Applications of cryptography

Abstract

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.

Problem

• Covering Radius Problem
• Binary Covering Radius Problem
• ℓ_p norm
• NP-hardness
• Π₂-hardness

Innovation

• Covering Radius Problem
• NP-hardness
• Binary GapCRP
• ℓ_p norms
• Approximation Hardness