Linear Space Streaming Lower Bounds for Approximating CSPs

arXiv:2106.13078v4 Announce Type: replace-cross Abstract: 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 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 ext