On Approximability of Satisfiable k-CSPs: IV
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3618260.3649610.pdf
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Author(s)
Bhangale, Amey
Khot, Subhash
Minzer, Dor
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Bhangale, Amey, Khot, Subhash and Minzer, Dor. 2024. "On Approximability of Satisfiable k-CSPs: IV."
Version
Final published version
Abstract
We prove a stability result for general 3-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if Σ,Γ and Φ are alphabets of constant size, and µ is a distribution over Σ×Γ×Φ satisfying: (1) the probability of each atom is at least Ω(1), (2) µ is pairwise connected, and (3) µ has no Abelian embeddings into (ℤ,+), then the following holds. Any triplets of 1-bounded functions f∶ Σn→ℂ, g∶ Γn→ℂ, h∶ Φn→ℂ satisfying
(x,y,z)∼ µ⊗ nf(x)g(y)h(z)≥
must arise from an Abelian group associated with the distribution µ. 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 f(x) = χ(σ(x1),…,σ(xn)) L (x), where σ∶ Σ → H is some map, χ∈ Ĥ⊗ n is a character, and L∶ Σn→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.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1145/3618260.3649610
