A Boolean constraint satisfaction problem (CSP), $\textsf{Max-CSP}(f)$, 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 consider the approximability of $\textsf{Max-CSP}(f)$ in the (dynamic) streaming setting, where constraints are inserted (and may also be deleted in the dynamic setting) one at a time. We completely characterize the approximability of all Boolean CSPs in the dynamic streaming setting. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of $\textsf{Max-CSP}(f)$ has a probabilistic dynamic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of $\textsf{Max-CSP}(f)$ requires $\Omega(\sqrt{n})$ space for probabilistic dynamic streaming algorithms. We also extend previously known results in the insertion-only setting to a wide variety of cases, and in particular the case of $k=2$ where we get a dichotomy and the case when the satisfying assignments of $f$ support a distribution on $\{-1,1\}^k$ with uniform marginals.
翻译:(CSP), $\ textsf{ max- CSP} (f) 是一个最大限制问题。 问题的例子包括对美元 Boolean 变量的$限制应用, 每个限制应用对从 美元变量及其否定中选择的 $k$ 字数适用限制。 目标是计算以美元为单位的 $( textsf{ max- CSP} (f) 的最大限制数量。 在 美元为 美元为 美元为 美元为可变 的 美元 中, 以美元为单位的 最大限制数量 。 在 $ (\ gamma,\ a) 美元为单位的 问题版本 : 美元= 1, 问题的例子包括$ 美元限制应用 $( m) 美元为美元 美元为美元 ; 以美元为单位的 美元为单位的 。 在这项工作中, 以美元为单位的 以美元为单位的 美元为单位的 ; 以美元为单位的 美元为单位的 。 以 美元为单位的 美元为单位的 。