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On Approximability of Satisfiable $k$-CSPs: VII

Amey Bhangale, Subhash Khot, Yang P. Liu, Dor Minzer

TL;DR

It is proved that if $\mu$ does not admit Abelian embeddings, and $f_i: \Sigma_i \to \mathbb{C}$ are 1-bounded functions, then there exists $d$ and $\delta>0$ depend only on $k, \alpha$ and $\varepsilon$.

Abstract

Let $Σ_1,\ldots,Σ_k$ be finite alphabets, and let $μ$ be a distribution over $Σ_1 \times \dots \times Σ_k$ in which the probability of each atom is at least $α$. We prove that if $μ$ does not admit Abelian embeddings, and $f_i: Σ_i \to \mathbb{C}$ are $1$-bounded functions (for $i=1,\ldots,k$) such that \[ \left|\mathbb{E}_{(x_1,\dots,x_k) \sim μ^{\otimes n}}\Big[f_1(x_1) \dots f_k(x_k)\Big]\right| \geq \varepsilon, \] then there exists $L\colon Σ_1^n\to\mathbb{C}$ of degree at most $d$ and $\|L\|_2\leq 1$ such that $|\langle f_1, L\rangle|\geq δ$, where $d$ and $δ>0$ depend only on $k, α$ and $\varepsilon$. This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.

On Approximability of Satisfiable $k$-CSPs: VII

TL;DR

It is proved that if does not admit Abelian embeddings, and are 1-bounded functions, then there exists and depend only on and .

Abstract

Let be finite alphabets, and let be a distribution over in which the probability of each atom is at least . We prove that if does not admit Abelian embeddings, and are -bounded functions (for ) such that \[ \left|\mathbb{E}_{(x_1,\dots,x_k) \sim μ^{\otimes n}}\Big[f_1(x_1) \dots f_k(x_k)\Big]\right| \geq \varepsilon, \] then there exists of degree at most and such that , where and depend only on and . This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.

Paper Structure

This paper contains 23 sections, 13 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Constraint Satisfaction Problems
  3. Towards Satisfiable CSPs
  4. Abelian Embeddings
  5. Our Contributions
  6. Main Result
  7. Implications to CSP approximability:
  8. Local Inverse Theorems under Milder Assumptions
  9. A Global Inverse Theorem
  10. Subsequent and Future Works
  11. Subsequent works:
  12. Future works:
  13. Preliminaries
  14. Notations:
  15. Noise Operators
  16. ...and 8 more sections

Key Result

Theorem 1

Let $k$ be a positive integer and let $\mu$ be a distribution over $\Sigma_1 \times \dots \times \Sigma_k$ that does not admit any Abelian embedding, and in which the probability of each atom is at least $\alpha$. Then, for every $\varepsilon > 0$ there is $\delta := \delta(\alpha,\varepsilon)>0$ su then $\mathsf{Stab}_{1-\delta}(f_i) \geqslant \delta$ for all $i = 1, 2, \dots, k$. Quantitatively,

Theorems & Definitions (31)

  • Definition 1.1
  • Conjecture 1.2
  • Theorem 1: Main theorem
  • Remark
  • Theorem 2
  • Definition 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 3
  • Remark
  • ...and 21 more