Stability Dichotomies for Boolean Constraint Satisfaction Problems

2026-09-26

AI Summary

This study addresses the long-standing absence of a unified theoretical framework for quantifying the algorithmic stability of Boolean constraint satisfaction problems (CSPs). By leveraging average sensitivity and Wasserstein distance in conjunction with the structural complexity theory of constraint languages, this work establishes, for the first time, precise connections between algorithmic stability and structural properties such as finite duality and bounded width. Specifically, it proves complete dichotomies for both stable solvability and stable approximability. The results demonstrate that finite duality and bounded width correspond to constant-level and logarithmic-level sensitivities, respectively, while all remaining cases exhibit linear lower bounds. Consequently, this research establishes a rigorous classification system for optimal sensitivity across different constraint languages, providing a comprehensive characterization of when Boolean CSPs admit stable algorithms.

Abstract

We study the stability of Boolean constraint satisfaction problems (CSPs) through the notion of average sensitivity (Varma and Yoshida, SODA 2021; SICOMP 2023). It measures the expected $1$-Wasserstein distance between an algorithm's output distributions before and after the deletion of a uniformly chosen constraint, using the unnormalized Hamming metric. We establish two dichotomies for every finite Boolean constraint language $\Gamma$, where $n\geq 2$ denotes the number of variables in an instance. For stable solvability, exactly one of the following holds: $\bullet$ either there is an algorithm that solves $\mathrm{CSP}(\Gamma)$ and has average sensitivity $O_{\Gamma}(1)$ for all satisfiable instances; $\bullet$ or every algorithm that solves $\mathrm{CSP}(\Gamma)$ has average sensitivity $\Omega_{\Gamma}(n)$ on satisfiable instances of arbitrarily large $n$. The first alternative holds if and only if $\Gamma$ has finite duality: unsatisfiability can be witnessed on a bounded number of variables. For stable approximability, where a $(1-\varepsilon)$-approximation violates at most an $\varepsilon$-fraction of the constraints in expectation, exactly one of the following holds: $\bullet$ either for every $\varepsilon\in(0,1]$, there is an algorithm that $(1-\varepsilon)$-approximates $\mathrm{CSP}(\Gamma)$ with average sensitivity $O_{\Gamma}(\varepsilon^{-1}\log n)$ for all satisfiable instances; $\bullet$ or there exists $\varepsilon_{\Gamma}\in(0,1]$ such that every algorithm that $(1-\varepsilon_{\Gamma})$-approximates $\mathrm{CSP}(\Gamma)$ has average sensitivity $\Omega_{\Gamma}(n)$ on satisfiable instances of arbitrarily large $n$. The first alternative holds if and only if $\Gamma$ has bounded width: local consistency checks on bounded sets of variables detect unsatisfiability.

Problem

• Boolean constraint satisfaction problems
• average sensitivity
• stability dichotomies
• stable solvability
• stable approximability

Innovation

• Boolean CSPs
• Average Sensitivity
• Stability Dichotomies
• Finite Duality
• Bounded Width