We study the single-pass streaming complexity of deciding satisfiability of Constraint Satisfaction Problems (CSPs). A CSP is specified by a constraint language $Γ$, that is, a finite set of $k$-ary relations over the domain $[q] = \{0, \dots, q-1\}$. An instance of $\mathsf{CSP}(Γ)$ consists of $m$ constraints over $n$ variables $x_1, \ldots, x_n$ taking values in $[q]$. Each constraint $C_i$ is of the form $\{R_i,(x_{i_1} + λ_{i_1}, \ldots, x_{i_k} + λ_{i_k})\}$, where $R_i \in Γ$ and $λ_{i_1}, \ldots, λ_{i_k} \in [q]$ are constants; it is satisfied if and only if $(x_{i_1} + λ_{i_1}, \ldots, x_{i_k} + λ_{i_k}) \in R_i$, where addition is modulo $q$. In the streaming model, constraints arrive one by one, and the goal is to determine, using minimum memory, whether there exists an assignment satisfying all constraints. For $k$-SAT, Vu (TCS 2024) proves an optimal $Ω(n^k)$ space lower bound, while for general CSPs, Chou, Golovnev, Sudan, and Velusamy (JACM 2024) establish an $Ω(n)$ lower bound; a complete characterization has remained open. We close this gap by showing that the single-pass streaming space complexity of $\mathsf{CSP}(Γ)$ is precisely governed by its non-redundancy, a structural parameter introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020). The non-redundancy $\mathsf{NRD}_n(Γ)$ is the maximum number of constraints over $n$ variables such that every constraint $C$ is non-redundant, i.e., there exists an assignment satisfying all constraints except $C$. We prove that the single-pass streaming complexity of $\mathsf{CSP}(Γ)$ is characterized, up to a logarithmic factor, by $\mathsf{NRD}_n(Γ)$.

翻译：我们研究单遍流式模型下判定约束满足问题（CSP）可满足性的空间复杂度。CSP由约束语言$\Gamma$定义，即定义在域$[q]=\{0,\dots,q-1\}$上的一组有限个$k$元关系。$\mathsf{CSP}(\Gamma)$的一个实例包含$m个约束$，这些约束作用于取值为$[q]$的$n$个变量$x_1,\ldots,x_n$。每个约束$C_i$的形式为$\{R_i,(x_{i_1}+\lambda_{i_1},\ldots,x_{i_k}+\lambda_{i_k})\}$，其中$R_i\in\Gamma$且$\lambda_{i_1},\ldots,\lambda_{i_k}\in[q]$为常数；该约束成立当且仅当$(x_{i_1}+\lambda_{i_1},\ldots,x_{i_k}+\lambda_{i_k})\in R_i$（加法为模$q$运算）。在流式模型中，约束逐个到达，目标是在最小内存使用下判断是否存在满足所有约束的赋值。对于$k$-SAT问题，Vu（TCS 2024）证明了最优的$\Omega(n^k)$空间下界，而针对一般CSP，Chou、Golovnev、Sudan和Velusamy（JACM 2024）建立了$\Omega(n)$下界；完整刻画问题此前尚未解决。我们通过证明$\mathsf{CSP}(\Gamma)$的单遍流式空间复杂度精确由其非冗余性（由Bessiere、Carbonnel和Katsirelos（AAAI 2020）引入的结构参数）决定来填补这一空白。非冗余性$\mathsf{NRD}_n(\Gamma)$定义为包含$n$个变量的实例中最大约束数量，使得每个约束$C$都是非冗余的，即存在一个满足除$C$外所有约束的赋值。我们证明，在忽略对数因子的情况下，$\mathsf{CSP}(\Gamma)$的单遍流式复杂度由$\mathsf{NRD}_n(\Gamma)$刻画。