布尔约束满足问题的稳定性二分法

中文摘要 AI 辅助

我们通过平均敏感度（Varma 和 Yoshida，SODA 2021；SICOMP 2023）的概念研究布尔约束满足问题（CSPs）的稳定性。它衡量在删除一个均匀选择的约束后，算法输出分布之间的期望 $1$-Wasserstein 距离，使用未归一化的汉明度量。我们为每个有限布尔约束语言 $\Gamma$ 建立两个二分法，其中 $n\geq 2$ 表示实例中的变量数量。对于稳定可解性，以下恰好之一成立：$\bullet$ 要么存在一个算法求解 $\mathrm{CSP}(\Gamma)$，并且对所有可满足实例具有平均敏感度 $O_{\Gamma}(1)$；$\bullet$ 要么每个求解 $\mathrm{CSP}(\Gamma)$ 的算法在任意大 $n$ 的可满足实例上具有平均敏感度 $\Omega_{\Gamma}(n)$。第一个替代成立当且仅当 $\Gamma$ 具有有限对偶性：不可满足性可以在有界数量的变量上被见证。对于稳定可近似性，其中 $(1-\varepsilon)$-近似在期望中违反至多 $\varepsilon$ 比例的约束，以下恰好之一成立：$\bullet$ 要么对于每个 $\varepsilon\in(0,1]$，存在一个算法对所有可满足实例以平均敏感度 $O_{\Gamma}(\varepsilon^{-1}\log n)$ 进行 $(1-\varepsilon)$-近似 $\mathrm{CSP}(\Gamma)$；$\bullet$ 要么存在 $\varepsilon_{\Gamma}\in(0,1]$ 使得每个 $(1-\varepsilon_{\Gamma})$-近似 $\mathrm{CSP}(\Gamma)$ 的算法在任意大 $n$ 的可满足实例上具有平均敏感度 $\Omega_{\Gamma}(n)$。第一个替代成立当且仅当 $\Gamma$ 具有有界宽度：对有界变量集的局部一致性检查能检测不可满足性。

英文摘要

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 $Γ$, 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 $\mathrm{CSP}(Γ)$ and has average sensitivity $O_Γ(1)$ for all satisfiable instances; $\bullet$ or every algorithm that solves $\mathrm{CSP}(Γ)$ has average sensitivity $Ω_Γ(n)$ on satisfiable instances of arbitrarily large $n$. The first alternative holds if and only if $Γ$ 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 $\mathrm{CSP}(Γ)$ with average sensitivity $O_Γ(\varepsilon^{-1}\log n)$ for all satisfiable instances; $\bullet$ or there exists $\varepsilon_Γ\in(0,1]$ such that every algorithm that $(1-\varepsilon_Γ)$-approximates $\mathrm{CSP}(Γ)$ has average sensitivity $Ω_Γ(n)$ on satisfiable instances of arbitrarily large $n$. The first alternative holds if and only if $Γ$ has bounded width: local consistency checks on bounded sets of variables detect unsatisfiability.

发表机构

• National Institute of Informatics（信息学国立研究所）

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