A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ 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 streaming algorithms and give a dichotomy result in the dynamic setting, where constraints can be inserted or deleted. Specifically, for every family ${\cal F}$ and every $\beta < \gamma$, we show that either the approximation problem is solvable with polylogarithmic space in the dynamic setting, or not solvable with $o(\sqrt{n})$ space. We also establish tight inapproximability results for a broad subclass in the streaming insertion-only setting. Our work builds on, and significantly extends previous work by the authors who consider the special case of Boolean variables ($q=2$), singleton families ($|{\cal F}| = 1$) and where constraints may be placed on variables or their negations. Our framework extends non-trivially the previous work allowing us to appeal to richer norm estimation algorithms to get our algorithmic results. For our negative results we introduce new variants of the communication problems studied in the previous work, build new reductions for these problems, and extend the technical parts of previous works.

翻译：限制满意度问题 (CSP), 最大 Max- CSP$ (xcal F}) 。 限制满意度问题由一系列有限的限制来指定 $[cal F} \ subseteq {q]\\ k\ k至 Q 0. 1, q美元和 美元。 美元变量问题的例子来自 $_ cal F} 至 美元变量的次序列。 目标是找到一个符合最大限制数量的变量的指派 。 在 $( gamma,\ beta) $ 的有限限制 中, 美元 美元 = = Qseteq = = q 美元 美元 。 目标在于区分以下几个例子: 将限制的至少 $\ gamma 部分应用到, 美元 来满足这些变量的后继 。 在这项工作中, 我们考虑这一问题的匹配性, 以 美元 。 美元 以 美元 美元 驱动算法 和 直方 表示 直方 的直方 的 的 的 。