A Subsampling Theorem for Constraint Satisfaction Problems with Large Arity

2026-07-31

AI Summary

This study addresses the problem of efficiently subsampling high-arity constraint satisfaction problems (CSPs) while preserving their optimal value up to a small approximation error. The authors propose a randomized subsampling method that selects only a small subset of variables yet accurately approximates the optimum of the original CSP. This work establishes the first subsampling theorem for CSPs of arbitrary arity $k$, achieving sample complexity polynomial in $k$ and the error parameter $\varepsilon$, and polylogarithmic in the alphabet size $q$. By integrating techniques from probabilistic analysis, property testing, and computational complexity theory, the approach not only provides a crucial ingredient in proving $\mathrm{AM}(\mathrm{poly}) = \mathrm{AM}$, but also yields the first one-sided CSP satisfiability tester with significantly improved sample complexity over prior results.

Abstract

Subsampling theorems for constraint satisfaction problems (CSPs) guarantee that the value of the CSP is approximately preserved after restricting it to small random subsets of variables. We provide the first subsampling theorem for CSPs, which requires a sample size that is polynomial in the arity $k$ and error $\varepsilon$, and polylogarithmic in the alphabet size $q$. This improves upon the subsampling theorem of Barak, Hardt, Holenstein, and Steurer (SODA '11), which achieves a polynomial dependency on $\varepsilon$ and polylogarithmic in $q$ only in the constant-arity regime. Our subsampling theorem has applications in interactive proofs and property testing. In interactive proofs, it provides a key missing ingredient for the proof of Aaronson, Impagliazzo, and Moshkovitz (CCC '14) that $\textsf{AM}(\textsf{poly})=\textsf{AM}$ (where $\textsf{AM}(k)$ is the class of languages decidable by Arthur-Merlin protocols with $k$ non-communicating Merlins with independent questions). In property testing, it yields the first one-sided tester for satisfiability with sample size polynomial in the arity $k$ and the error $\varepsilon^{-1}$, and polylogarithmic in the alphabet size $q$.

Problem

• constraint satisfaction problems
• subsampling
• large arity
• interactive proofs
• property testing

Innovation

• subsampling theorem
• constraint satisfaction problems
• large arity
• interactive proofs
• property testing