A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal is to compute the maximum number of constraints that can be satisfied by a Boolean assignment to the $n$~variables. In the $(\gamma,\beta)$-approximation version of the problem for parameters $\gamma \geq \beta \in [0,1]$, 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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic streaming algorithm using $O(\log n)$ space, or (2) for every $\varepsilon > 0$ the $(\gamma-\varepsilon,\beta+\varepsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic streaming algorithms. Previously such a separation was known only for $k=2$. We stress that for $k=2$, there are only finitely many distinct problems to consider. Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17], [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to explore biases. Our negative results combine the Fourier analytic methods of [Kapralov-Khanna-Sudan SODA'15], which we extend to a wider class of CSPs, with a rich collection of reductions among communication complexity problems that lie at the heart of the negative results.

中文翻译：

---

布尔CSP的流近似的分类

布尔约束满足问题（CSP），Max-CSP $（f）$，是由约束$ f：\ {-1,1 \} ^ k \ to \ {0,1 \} $指定的最大化问题。问题的一个实例包括对$ n $布尔变量的$ m $约束应用程序，其中每个约束应用程序都将约束应用于从$ n $变量及其取反中选择的$ k $文字。目的是计算通过对$ n $〜变量进行布尔分配可以满足的最大约束数量。在参数$ \ gamma \ geq \ beta \ in [0,1] $的问题的近似（$（\ gamma，\ beta）$）版本中，目标是区分至少$ \ gamma $分数的实例可以从最多满足约束的$ \ beta $分数的实例中满足约束条件。在这项工作中，我们完全描述了流模型中所有布尔CSP的近似性。具体来说，给定$ f $，$ \ gamma $和$ \ beta $，我们证明（1）Max-CSP $（f）$的$（\ gamma，\ beta）$逼近版本具有概率流算法使用$ O（\ log n）$空间，或者对于每个$ \ varepsilon> 0 $使用（2）$（\ gamma- \ varepsilon，\ beta + \ varepsilon）$近似版本的Max-CSP $（f）$需要$ \ Omega（\ sqrt {n}）$空间用于概率流算法。以前，这种分离仅在$ k = 2 $时才知道。我们强调，对于$ k = 2 $，只有有限的许多不同的问题需要考虑。我们的积极结果表明，[Guruswami-Velingker-Velusamy APPROX'17]，[Chou-Golovnev-Velusamy FOCS'20]以前使用的基于偏差的算法通过提供系统性的探索偏差的方法，具有更广泛的适用性。