We show a dichotomy result for $p$-pass streaming algorithms for all CSPs and for up to polynomially many passes. More precisely, we prove that for any arity parameter $k$, finite alphabet $Σ$, collection $\mathcal{F}$ of $k$-ary predicates over $Σ$ and any $c\in (0,1)$, there exists $0<s\leq c$ such that: 1. For any $\varepsilon>0$ there is a constant pass, $O_{\varepsilon}(\log n)$-space randomized streaming algorithm solving the promise problem $\text{MaxCSP}(\mathcal{F})[c,s-\varepsilon]$. That is, the algorithm accepts inputs with value at least $c$ with probability at least $2/3$, and rejects inputs with value at most $s-\varepsilon$ with probability at least $2/3$. 2. For all $\varepsilon>0$, any $p$-pass (even randomized) streaming algorithm that solves the promise problem $\text{MaxCSP}(\mathcal{F})[c,s+\varepsilon]$ must use $Ω_{\varepsilon}(n^{1/3}/p)$ space. Our approximation algorithm is based on a certain linear-programming relaxation of the CSP and on a distributed algorithm that approximates its value. This part builds on the works [Yoshida, STOC 2011] and [Saxena, Singer, Sudan, Velusamy, SODA 2025]. For our hardness result we show how to translate an integrality gap of the linear program into a family of hard instances, which we then analyze via studying a related communication complexity problem. That analysis is based on discrete Fourier analysis and builds on a prior work of the authors and on the work [Chou, Golovnev, Sudan, Velusamy, J.ACM 2024].

翻译：我们展示了所有CSP及至多多项式遍数下$p$遍流式算法的二分类结果。更精确地，我们证明：对任意元数参数$k$、有限字母表$\Sigma$、$\Sigma$上$k$元谓词集合$\mathcal{F}$以及任意$c\in (0,1)$，存在$0<s\leq c$使得：1. 对任意$\varepsilon>0$，存在常数遍数、$O_{\varepsilon}(\log n)$空间的随机流式算法，可解决承诺问题$\text{MaxCSP}(\mathcal{F})[c,s-\varepsilon]$。即该算法以至少$2/3$的概率接受值至少为$c$的输入，并以至少$2/3$的概率拒绝值至多为$s-\varepsilon$的输入。2. 对所有$\varepsilon>0$，任何解决承诺问题$\text{MaxCSP}(\mathcal{F})[c,s+\varepsilon]$的$p$遍（甚至随机化）流式算法必须使用$\Omega_{\varepsilon}(n^{1/3}/p)$空间。我们的近似算法基于CSP的特定线性规划松弛以及逼近其值的分布式算法。该部分建立在[Yoshida, STOC 2011]和[Saxena, Singer, Sudan, Velusamy, SODA 2025]的工作基础上。在困难性结果中，我们展示了如何将线性规划的整数间隙转化为一族困难实例，并通过研究相关通信复杂度问题进行分析。该分析基于离散傅里叶分析，并建立在作者前期工作与[Chou, Golovnev, Sudan, Velusamy, J.ACM 2024]的基础上。