We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $Ω(n)$ space even on instances with $O(n)$ constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires $Ω(n)$ space. The key technical core is an optimal, $q^{-(k-1)}$-inapproximability for the Max $k$-LIN-$\bmod\; q$ problem, which is the Max CSP problem where every constraint is given by a system of $k-1$ linear equations $\bmod\; q$ over $k$ variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max $k$-LIN-$\bmod\; q$ with ${k=q=2}$. For general CSPs in the streaming setting, prior results only yielded $Ω(\sqrt{n})$ space bounds. In particular no linear space lower bound was known for an approximation factor less than $1/2$ for any CSP. Extending the work of Kapralov and Krachun to Max $k$-LIN-$\bmod\; q$ to $k>2$ and $q>2$ (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work.

翻译：我们考虑流式环境下约束满足问题的可近似性。对于定义在变量集$\{0,\ldots,q-1\}$上的$n$个变量的每个约束满足问题（CSP），我们证明：即使实例仅包含$O(n)$个约束，若要在平凡近似比的基础上改进$q$倍，则仍需$\Omega(n)$空间。此外，我们识别出一类广泛子问题，任何超越平凡近似比的改进均需$\Omega(n)$空间。关键技术核心是Max $k$-LIN-$\bmod\; q$问题的最优$q^{-(k-1)}$不可近似性——该最大CSP问题中每个约束由$k$个变量上的$k-1$个线性方程模$q$系统定义。本研究建立在Kapralov与Krachun（STOC 2019）的突破性工作基础上并加以扩展，他们证明了图中MaxCut问题的任何非平凡近似均具有线性下界。MaxCut大致对应$k=q=2$时的Max $k$-LIN-$\bmod\; q$问题。此前流式CSP的结果仅能导出$\Omega(\sqrt{n})$空间界，且对于任意CSP，近似比小于$1/2$时均无线线性空间下界已知。将Kapralov与Krachun的工作推广至$k>2$且$q>2$的Max $k$-LIN-$\bmod\; q$问题（同时获得最优不可近似性结果）是本文的主要技术贡献。每项扩展均带来非平凡技术挑战，我们已在研究中克服。