# Approximability of all Boolean CSPs in the dynamic streaming setting

A Boolean constraint satisfaction problem (CSP), Max-CSP(f), is a maximization problem specified by a constraint f:{-1,1}^k→{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 (γ,β)-approximation version of the problem for parameters γ≥β∈ [0,1], the goal is to distinguish instances where at least γ fraction of the constraints can be satisfied from instances where at most β 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, γ and β we show that either (1) the (γ,β)-approximation version of Max-CSP(f) has a probabilistic dynamic streaming algorithm using O(log n) space, or (2) for every ε > 0 the (γ-ε,β+ε)-approximation version of Max-CSP(f) requires Ω(√(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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