可满足 $k$-CSP 的可近似性：VI
计算复杂性
2024-11-25 v1 组合数学
摘要
我们为一般的 3 3 3 μ \mu μ Σ × Γ × Φ \Sigma \times \Gamma \times \Phi Σ × Γ × Φ μ x y \mu_{xy} μ x y μ x z \mu_{xz} μ x z μ y z \mu_{yz} μ y z f : Σ n → C f: \Sigma^n \to \mathbb{C} f : Σ n → C g : Γ n → C g: \Gamma^n \to \mathbb{C} g : Γ n → C h : Φ n → C h: \Phi^n \to \mathbb{C} h : Φ n → C 1 1 1 ∣ E ( x , y , z ) ∼ μ ⊗ n [ f ( x ) g ( y ) h ( z ) ] ∣ ≥ ε . \left|\mathbb{E}_{(x,y,z) \sim \mu^{\otimes n}}[f(x)g(y)h(z)]\right| \geq \varepsilon. E ( x , y , z ) ∼ μ ⊗ n [ f ( x ) g ( y ) h ( z )] ≥ ε . δ : = exp ( − ε − O μ ( 1 ) ) \delta := \exp(-\varepsilon^{-O_{\mu}(1)}) δ := exp ( − ε − O μ ( 1 ) ) δ \delta δ f f f δ n \delta n δ n δ \delta δ f f f δ n \delta n δ n δ \delta δ f f f L ⋅ P L \cdot P L ⋅ P 2 − poly ( log ( 1 / δ ) ) 2^{-\text{poly}(\log(1/\delta))} 2 − poly ( l o g ( 1/ δ )) L L L poly ( 1 / δ ) \text{poly}(1/\delta) poly ( 1/ δ ) ∥ L ∥ 2 ≤ 1 \|L\|_2 \leq 1 ∥ L ∥ 2 ≤ 1 P P P Σ \Sigma Σ S ⊆ Σ × Σ × Σ S \subseteq \Sigma \times \Sigma \times \Sigma S ⊆ Σ × Σ × Σ x ∈ S x \in S x ∈ S ( x , x , x ) ∈ S (x, x, x) \in S ( x , x , x ) ∈ S μ x y \mu_{xy} μ x y μ x z \mu_{xz} μ x z μ y z \mu_{yz} μ y z A ⊆ Σ n A \subseteq \Sigma^n A ⊆ Σ n ∣ Σ ∣ − n ∣ A ∣ ≥ Ω ( ( log log log n ) − c ) |\Sigma|^{-n}|A| \geq \Omega((\log \log \log n)^{-c}) ∣Σ ∣ − n ∣ A ∣ ≥ Ω (( log log log n ) − c ) x , y , z ∈ A x, y, z \in A x , y , z ∈ A i i i ( x i , y i , z i ) ∈ S (x_i, y_i, z_i) \in S ( x i , y i , z i ) ∈ S 3 3 3
引用
@article{arxiv.2411.15133,
title = {On Approximability of Satisfiable $k$-CSPs: VI},
author = {Amey Bhangale and Subhash Khot and Yang P. Liu and Dor Minzer},
journal= {arXiv preprint arXiv:2411.15133},
year = {2024}
}