A constraint satisfaction problem (CSP), $\textsf{Max-CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from $\mathcal{F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(\gamma,\beta)$-approximation version of the problem for parameters $0 \leq \beta < \gamma \leq 1$, the goal is to distinguish instances where at least $\gamma$ fraction of the constraints can be satisfied from instances where at most $\beta$ fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family $\mathcal{F}$ and every $\beta < \gamma$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of $q=k=2$, where we get a dichotomy, and the case when the satisfying assignments of the constraints of $\mathcal{F}$ support a distribution on $[q]^k$ with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables $q=2$, binary constraints $k=2$, singleton families $|\mathcal{F}|=1$ and only considered the setting where constraints are placed on literals rather than variables.

翻译：约束满足问题（CSP），即$\textsf{Max-CSP}(\mathcal{F})$，由有限约束集$\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$定义，其中$q$和$k$为正整数。该问题的$n$变量实例由从$\mathcal{F$}中选取的约束应用于$n$变量子序列的$m$次操作构成，目标是为变量找到满足最多约束的赋值。在参数$0 \leq \beta < \gamma \leq 1$的$(\gamma,\beta)$-近似版本中，目标是区分至少有$\gamma$比例的约束可满足的实例与最多仅有$\beta$比例的约束可满足的实例。本文从素描算法角度研究该问题的近似性，并给出二分性结论。具体而言，对每个族$\mathcal{F}$及每个$\beta < \gamma$，我们证明：要么存在线性素描算法可用多对数空间求解该问题，要么该问题无法被任何素描算法用$o(\sqrt{n})$空间求解。特别地，我们为无穷多个约束满足问题给出了使用多对数空间的非平凡近似算法。我们还将通用流算法的已知下界推广至广泛问题类别，尤其在$q=k=2$情形（此时得到二分性）以及$\mathcal{F}$中约束的可满足赋值在$[q]^k$上支持均匀边缘分布的情形。此前，除零散实例外，唯一被系统分析的CSP类别仅涉及布尔变量$q=2$、二元约束$k=2$、单一族$|\mathcal{F}|=1$，且仅考虑约束作用在文字而非变量的设定。