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

Huck Bennett, Peter Ly

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.

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

Abstract

We study the hardness of the -approximate decisional Covering Radius Problem on lattices in the norm (-). Specifically, we prove that there is an explicit function , with for and , such that for any constant , - is -hard. This shows the first hardness of for explicit . Work of Haviv and Regev (CCC, 2006 and CJTCS, 2012) previously showed -hardness of approximation for for all sufficiently large (but non-explicit) finite and for . In fact, our hardness results hold for a variant of called the Binary Covering Radius Problem (), which trivially reduces to both and the decisional Linear Discrepancy Problem () in any norm in an approximation-preserving way. We also show -hardness of - in the norm for any constant . Our work extends and heavily uses the work of Manurangsi (IPL, 2021), which showed -hardness of - in the norm.
Paper Structure (21 sections, 18 theorems, 52 equations, 1 figure)

This paper contains 21 sections, 18 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. The Haviv-Regev hardness reduction.
  5. Combinatorial and linear discrepancy.
  6. Technical Overview
  7. Open Questions
  8. Acknowledgements
  9. Preliminaries
  10. Covering Radius and Linear Discrepancy
  11. Covering radius versus linear discrepancy.
  12. Covering Radius and Linear Discrepancy as Computational Problems
  13. Constraint Satisfaction Problems
  14. Reduction from NAE3SAT to BinaryGapCRP
  15. The Reduction
  16. ...and 6 more sections

Key Result

Theorem 1.1

For all $p > p_0 := 35.310188\ldots$ there exists an explicit value $\gamma(p) > 1$, specified in eq:np-hardness-factor, such that for any constant $\eps > 0$, $(\gamma(p) - \eps)$-$\problem{BinGapCRP}_p$ is $\NP$-hard. Furthermore, the values $\gamma(p)$ satisfy $\lim_{p \to \infty} \gamma(p) = 9/8

Figures (1)

  • Figure 1: A linear-log plot of the approximation factor $\gamma = \gamma(p)$ for which we show $\NP$-hardness of $(\gamma - \eps)$-$\problem{BinGapCRP}$ in the $\ell_p$ norm (in blue; see \ref{['thm:intro-np-hardness']}). The function $\gamma(p)$, specified in \ref{['eq:np-hardness-factor']}, is greater than $1$ for all $p > p_0 = 35.310188...$ and satisfies $\lim_{p \to \infty} \gamma(p) = 9/8$. (The blue dot is at the point $(p, \gamma) = (p_0, 1)$, where $p_0$ is the unique value of $p$ such that $\gamma(p) = 1$. The horizontal asymptote of $9/8$ is shown in red.) This factor of $9/8$ also appears in our $\Pi_2$-hardness result for $\problem{BinGapCRP}_{\infty}$ (\ref{['thm:intro-pi2-hardness']}) and in the work of Manurangsi manurangsi21 on $\problem{LinDisc}$, which is the foundation of our work.

Theorems & Definitions (42)

  • Theorem 1.1: $\NP$-hardness of approximation for $\problem{BinGapCRP}_p$
  • Theorem 1.2: $\Pi_2$-hardness of approximation for $\problem{BinGapCRP}_{\infty}$
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.6: Constraint Satisfaction Problems (CSP)
  • Definition 2.7: $(\delta, \eps)$-$\text{NAE-E$k$-SAT}$ problem
  • Theorem 2.8
  • Lemma 3.1
  • ...and 32 more