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 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 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 Max-CSP$(f)$ has a probabilistic dynamic 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 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.
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