For an arbitrary family of predicates $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ and any $ε> 0$, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP$({\mathcal{F}})$ with at most $β+ε$ fraction of satisfiable constraints from instances of with at least $γ-ε$ fraction of satisfiable constraints, whenever Max-CSP$({\mathcal{F}})$ admits a $(γ,β)$-integrality gap instance for the basic LP. This subsumes the linear-space lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC 2022), which applies only to a special subclass of CSPs with linear-algebraic structure. (Their result itself generalizes work of Kapralov and Krachun (STOC 2019) for Max-CUT.) Our approach identifies the right ``analytic'' analogues of previously-used linear-algebraic conditions; this yields substantial simplifications while capturing a much larger class of problems. Our lower bound is essentially optimal for single-pass streaming, since: (1) All CSPs admit $(1-ε)$-approximations in quasilinear space, and (2) sublinear-space streaming algorithms can simulate the LP (on bounded-degree instances), giving approximation algorithms when integrality gap instances do not exist. The starting point for our lower bound is a reduction from a "distributional implicit hidden partition'' problem defined by Fei, Minzer, and Wang (STOC 2026) in the context of multi-pass streaming. Our result is an analogue of theirs in the single-pass setting, where we obtain a much stronger (and tight) space lower bound.

翻译：对于任意谓词族 $\mathcal{F} \subseteq \{0,1\}^{[q]^k}$ 及任意 $ε> 0$，我们证明了一个针对间隙判定问题的单遍、线性空间流式下界：该问题需区分 Max-CSP$({\mathcal{F}})$ 实例中可满足约束比例至多为 $β+ε$ 的实例与至少为 $γ-ε$ 的实例，前提是 Max-CSP$({\mathcal{F}})$ 对基本线性规划（LP）具有 $(γ,β)$-积分间隙实例。此结果涵盖了 Chou、Golovnev、Sudan、Velingker 和 Velusamy（STOC 2022）的线性空间下界，后者仅适用于具有线性代数结构的特殊 CSP 子类（其成果本身推广了 Kapralov 和 Krachun（STOC 2019）关于 Max-CUT 的工作）。我们的方法识别出先前所用线性代数条件的正确“解析”类比，从而在简化证明的同时覆盖了更广泛的问题类别。该下界对于单遍流式算法本质上是紧的，原因如下：(1) 所有 CSP 均可在拟线性空间内实现 $(1-ε)$-近似；(2) 次线性空间流式算法可模拟 LP（在有界度实例上），从而在无积分间隙实例时给出近似算法。我们下界的起点源于 Fei、Minzer 和 Wang（STOC 2026）在多遍流式背景下定义的“分布隐式隐划分”问题的归约。我们的结果是在单遍模式下的类比，并由此得到了一个更强的（且紧的）空间下界。