Linear Space Streaming Lower Bounds for Approximating CSPs

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 $\Omega(n)$ 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 $\Omega(n)$ space. The key technical core is an optimal, $q^{-(k-1)}$-inapproximability for the case where every constraint is given by a system of $k-1$ linear equations $\bmod\; q$ over $k$ variables. Prior to our work, no such hardness was known for an approximation factor less than $1/2$ for any CSP. Our work builds on and extends the work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the max cut in graphs. This corresponds roughly to the case of Max $k$-LIN-$\bmod\; q$ with $k=q=2$. Each one of the extensions provides non-trivial technical challenges that we overcome in this work.