In a streaming constraint satisfaction problem (streaming CSP), a $p$-pass algorithm receives the constraints of an instance sequentially, making $p$ passes over the input in a fixed order, with the goal of approximating the maximum fraction of satisfiable constraints. We show near optimal space lower bounds for streaming CSPs, improving upon prior works. (1) Fei, Minzer and Wang (\textit{STOC 2026}) showed that for any CSP, the basic linear program defines a threshold $α_{\mathrm{LP}}\in [0,1]$ such that, for any $\varepsilon > 0$, an $(α_{\mathrm{LP}} - \varepsilon)$-approximation can be achieved using constant passes and polylogarithmic space, whereas achieving $(α_{\mathrm{LP}} + \varepsilon)$-approximation requires $Ω(n^{1/3}/p)$ space. We improve this lower bound to $Ω(\sqrt{n}/p)$, which is nearly tight for a gap version of the problem. (2) For $p=o(\log n)$, we further strengthen the lower bound to $Ω(n\cdot2^{-O_{\varepsilon}(p)})$. Combined with existing algorithmic results, this shows that $α_{\mathrm{LP}}$ is not only the limit of multi-pass polylogarithmic-space algorithms, but also the limit of single-pass sublinear-space algorithms on bounded-degree instances. (3) For certain CSPs, we show that there exists $α< 1$ such that achieving an $α$-approximation requires $Ω(n/p)$ space. Our proofs are Fourier analytic, building on the techniques of Fei, Minzer and Wang (\textit{STOC 2026}) and the Fourier-$\ell_1$-based lower bound method of Kapralov and Krachun (\textit{STOC 2019}).

翻译：在一个流式约束满足问题（streaming CSP）中，一个$p$遍算法按固定顺序依次接收实例的约束，对输入进行$p$遍扫描，目标是近似计算可满足约束的最大比例。我们改进了先前的工作，给出了流式CSP的近最优空间下界。(1) Fei、Minzer和Wang（《STOC 2026》）证明，对于任何CSP，基本线性程序定义了一个阈值$α_{\mathrm{LP}}\in [0,1]$，使得对于任意$\varepsilon > 0$，使用常数遍和多对数空间可实现$(α_{\mathrm{LP}} - \varepsilon)$近似，而达到$(α_{\mathrm{LP}} + \varepsilon)$近似需要$Ω(n^{1/3}/p)$空间。我们将此下界改进为$Ω(\sqrt{n}/p)$，这对于问题的间隙版本几乎是紧的。(2) 对于$p=o(\log n)$，我们进一步将下界加强为$Ω(n\cdot2^{-O_{\varepsilon}(p)})$。结合现有算法结果，这表明$α_{\mathrm{LP}}$不仅是多遍多对数空间算法的极限，也是单遍次线性空间算法在有界度实例上的极限。(3) 对于某些CSP，我们证明存在$α< 1$，使得达到$α$近似需要$Ω(n/p)$空间。我们的证明基于傅里叶分析，构建于Fei、Minzer和Wang（《STOC 2026》）的技术以及Kapralov和Krachun（《STOC 2019》）的傅里叶-$\ell_1$-基下界方法之上。