Weizmann Logo
Electronic Colloquium on Computational Complexity

Under the auspices of the Computational Complexity Foundation (CCF)

Login | Register | Classic Style



TR23-112 | 30th July 2023 17:36

On Approximability of Satisfiable k-CSPs: IV


Authors: Amey Bhangale, Subhash Khot, Dor Minzer
Publication: 30th July 2023 17:40
Downloads: 187


We prove a stability result for general $3$-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if $\Sigma,\Gamma$ and $\Phi$ are alphabets of constant size, and $\mu$ is a pairwise connected distribution over $\Sigma\times\Gamma\times\Phi$ with no $(\mathbb{Z},+)$ embeddings in which the probability of each atom is $\Omega(1)$, then the following holds. Any triplets of $1$-bounded functions $f\colon \Sigma^n\to\mathbb{C}$, $g\colon \Gamma^n\to\mathbb{C}$, $h\colon \Phi^n\to\mathbb{C}$ satisfying
\left|\mathbb{E}_{(x,y,z)\sim \mu^{\otimes
n}}\big[f(x)g(y)h(z)\big]\right|\geq \varepsilon
must arise from an Abelian group associated with the distribution $\mu$. More specifically, we show that there is an Abelian group $(H,+)$ of constant size such that for any such $f,g$ and $h$, the function $f$ (and similarly $g$ and $h$) is correlated with a function of the form $\tilde{f}(x) = \chi(\sigma(x_1),\ldots,\sigma(x_n)) L (x)$, where $\sigma\colon \Sigma \to H$ is some map, $\chi\in \hat{H}^{\otimes n}$ is a character, and $L\colon
\Sigma^n\to\mathbb{C}$ is a low-degree function with bounded $2$-norm.

En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a ``restriction inverse theorem'' about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function. In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.

ISSN 1433-8092 | Imprint