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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.
A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:{-1,1}^kto{0,1}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint
We demonstrate a lower bound technique for linear decision lists, which are decision lists where the queries are arbitrary linear threshold functions. We use this technique to prove an explicit lower bound by showing that any linear decision list com
We develop a notion of {em inner rank} as a tool for obtaining lower bounds on the rank of matrix multiplication tensors. We use it to give a short proof that the border rank (and therefore rank) of the tensor associated with $ntimes n$ matrix multip
In the communication problem $mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x$ and $y$ in ${0,1}^n$ with the promise that $x eq y$. The last player to receive a message must output an index $i$ such that $x_i eq y_i$.
In the communication problem $mathbf{UR}$ (universal relation) [KRW95], Alice and Bob respectively receive $x, y in{0,1}^n$ with the promise that $x eq y$. The last player to receive a message must output an index $i$ such that $x_i eq y_i$. We prove