Characterizing Streaming Decidability of CSPs via Non-Redundancy

Abstract

We study the single-pass streaming complexity of deciding satisfiability of Constraint Satisfaction Problems (CSPs). A CSP is specified by a constraint language $\Gamma$, that is, a finite set of $k$-ary relations over the domain $[q] = \{0, \dots, q-1\}$. An instance of $\mathsf{CSP}(\Gamma)$ consists of $m$ constraints over $n$ variables $x_1, \ldots, x_n$ taking values in $[q]$. Each constraint $C_i$ is of the form $\{R_i,(x_{i_1} + \lambda_{i_1}, \ldots, x_{i_k} + \lambda_{i_k})\}$, where $R_i \in \Gamma$ and $\lambda_{i_1}, \ldots, \lambda_{i_k} \in [q]$ are constants; it is satisfied if and only if $(x_{i_1} + \lambda_{i_1}, \ldots, x_{i_k} + \lambda_{i_k}) \in R_i$, 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 $\Omega(n^k)$ space lower bound, while for general CSPs, Chou, Golovnev, Sudan, and Velusamy (JACM 2024) establish an $\Omega(n)$ lower bound; a complete characterization has remained open. We close this gap by showing that the single-pass streaming space complexity of $\mathsf{CSP}(\Gamma)$ is precisely governed by its non-redundancy, a structural parameter introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020). The non-redundancy $\mathsf{NRD}_n(\Gamma)$ 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 $\mathsf{CSP}(\Gamma)$ is characterized, up to a logarithmic factor, by $\mathsf{NRD}_n(\Gamma)$.