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On the (Classical and Quantum) Fine-Grained Complexity of Log-Approximate CVP and Max-Cut

Jeremy Ahrens Huang, Young Kun Ko, Chunhao Wang

TL;DR

It is shown that any sub-exponential time algorithm for theo(\sqrt{\log n}^{\frac{1}{p}})$-Approximate Closest Vector Problem in any finite $\ell_p$-norm implies a faster than the state-of-the-art sub-exponential time (classical or quantum) algorithm for Max-Cut.

Abstract

We show a linear sized reduction from the Maximum Cut Problem (Max-Cut) with completeness $1 - \varepsilon$ and soundness $1 - \varepsilon^{1/2}$ to the $γ$-Approximate Closest Vector Problem under any finite $\ell_p$-norm including $p = 2$. This reduction implies two headline results: (i) We show that any sub-exponential time (classical or quantum) algorithm for the $o(\sqrt{\log n}^{\frac{1}{p}})$-Approximate Closest Vector Problem in any finite $\ell_p$-norm implies a faster than the state-of-the-art (by Arora, Barak, and Steurer [\textit{Journal of the ACM}, 2015]) sub-exponential time (classical or quantum) algorithm for Max-Cut. This fills the gap between the results by Bennett, Golovnev, and Stephens-Davidowitz [\textit{FOCS} 2017] which had an almost optimal runtime lower bound but a very small approximation factor and the results by Dinur, Kindler, Raz, and Safra [\textit{Combinatorica}, 2003] which had an almost optimal approximation factor but small runtime lower bound, albeit using a different underlying hard problem; (ii) in combination with the classical results of Aggarwal and Kumar [\textit{FOCS} 2023] and our quantization of those results, there are no fine-grained reductions from $k$-SAT to Max-Cut with one-sided error, nor are there non-adaptive fine-grained (classical or quantum) reductions with two-sided error, unless the polynomial hierarchy collapses (or unless $\mathrm{NP} \subseteq \mathrm{pr} \text{-} \mathrm{QSZK}$ in the quantum case). The second result poses a significant barrier against proving the fine-grained complexity of Max-Cut using the Strong Exponential Time Hypothesis (or the Quantum Strong Exponential Time Hypothesis).

On the (Classical and Quantum) Fine-Grained Complexity of Log-Approximate CVP and Max-Cut

TL;DR

It is shown that any sub-exponential time algorithm for theo(\sqrt{\log n}^{\frac{1}{p}})\ell_p$-norm implies a faster than the state-of-the-art sub-exponential time (classical or quantum) algorithm for Max-Cut.

Abstract

We show a linear sized reduction from the Maximum Cut Problem (Max-Cut) with completeness and soundness to the -Approximate Closest Vector Problem under any finite -norm including . This reduction implies two headline results: (i) We show that any sub-exponential time (classical or quantum) algorithm for the -Approximate Closest Vector Problem in any finite -norm implies a faster than the state-of-the-art (by Arora, Barak, and Steurer [\textit{Journal of the ACM}, 2015]) sub-exponential time (classical or quantum) algorithm for Max-Cut. This fills the gap between the results by Bennett, Golovnev, and Stephens-Davidowitz [\textit{FOCS} 2017] which had an almost optimal runtime lower bound but a very small approximation factor and the results by Dinur, Kindler, Raz, and Safra [\textit{Combinatorica}, 2003] which had an almost optimal approximation factor but small runtime lower bound, albeit using a different underlying hard problem; (ii) in combination with the classical results of Aggarwal and Kumar [\textit{FOCS} 2023] and our quantization of those results, there are no fine-grained reductions from -SAT to Max-Cut with one-sided error, nor are there non-adaptive fine-grained (classical or quantum) reductions with two-sided error, unless the polynomial hierarchy collapses (or unless in the quantum case). The second result poses a significant barrier against proving the fine-grained complexity of Max-Cut using the Strong Exponential Time Hypothesis (or the Quantum Strong Exponential Time Hypothesis).

Paper Structure

This paper contains 42 sections, 29 theorems, 27 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Previous results
  3. Our results
  4. Approximate CVP is Max-Cut-hard
  5. Hardness of proving the hardness of Max-Cut
  6. Hardness of quantum hardness of CVP and Max-Cut
  7. Faster algorithms and quantum advantage for gap-Max-Cut
  8. Our reduction
  9. Construction
  10. Approximation factors
  11. Open questions
  12. Organization of this paper
  13. Acknowledgements
  14. Preliminaries
  15. Notation
  16. ...and 27 more sections

Key Result

Theorem 1.1

If there is an algorithm (classical or quantum) which decides $\gamma \text{-}$$\mathrm{CVP}^{\{0,1\}}_{p}$ with finite $p \geq 1$ in time $O (f (n, m))$ where $f (n, m) = \Omega (n + m)$, then for $\varepsilon, c \in (0, 1)$ such that $\varepsilon^{(c - 1) / p} \leq \gamma$ there is an algorithm (o

Figures (4)

  • Figure 1: (Color online.) A classical landscape for $\gamma \text{-}$$\mathrm{CVP}_2$.
  • Figure 2: Known upper and lower bounds for $(1 - \varepsilon, 1 - \varepsilon^c) \text{-} \mathrm{gap}$$\mathrm{Max}$-$\mathrm{Cut}$
  • Figure 3: components of the lattice basis along $\bm{e}_{2k}$ and $\bm{e}_{2k-1}$
  • Figure 4: Map of reductions from $k$-SAT to CVP$_2$ with edges labeled by reduction sizes.

Theorems & Definitions (67)

  • Theorem 1.1: Main theorem. Informal version of \ref{['thm:weighted-main-theorem']}
  • Conjecture 2.1: SETH
  • Conjecture 2.2: QSETH ACL+20
  • Conjecture 2.3: ETH IP99
  • Definition 2.4: Lattice, basis, coordinates, rank, and distance
  • Definition 2.5: The Closest Vector Problem
  • Definition 2.6: Binary Closest Vector Problem
  • Definition 2.7: Unique Label Cover Value
  • Definition 2.8: The Unique Label Problem
  • Remark 2.9
  • ...and 57 more