Near-Optimal Space Lower Bounds for Streaming CSPs

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 $\alpha_{\mathrm{LP}}\in [0,1]$ such that, for any $\varepsilon > 0$, an $(\alpha_{\mathrm{LP}} - \varepsilon)$-approximation can be achieved using constant passes and polylogarithmic space, whereas achieving $(\alpha_{\mathrm{LP}} + \varepsilon)$-approximation requires $\Omega(n^{1/3}/p)$ space. We improve this lower bound to $\Omega(\sqrt{n}/p)$, which is nearly tight for a gap version of the problem. (2) For $p=o(\log n)$, we further strengthen the lower bound to $\Omega(n\cdot2^{-O_{\varepsilon}(p)})$. Combined with existing algorithmic results, this shows that $\alpha_{\mathrm{LP}}$ 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 $\alpha < 1$ such that achieving an $\alpha$-approximation requires $\Omega(n/p)$ 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}).