Classification of the streaming approximability of Boolean CSPs

A Boolean constraint satisfaction problem (CSP), Max-CSP(f), is a maximization problem specified by a constraint f : {−1, 1} → {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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given f , γ and β we show that either (1) the (γ, β)-approximation version of Max-CSP(f) has a probabilistic 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 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 [GVV17], [CGV20] by giving a systematic way to explore biases. Our negative results combine the Fourier analytic methods of [KKS15], 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. ∗School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA. Supported by NSF awards CCF 1565264 and CNS 1618026. Email: chiningchou@g.harvard.edu. †Department of Computer Science, Georgetown University. Email: alexgolovnev@gmail.com. ‡School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA. Supported in part by a Simons Investigator Award and NSF Award CCF 1715187. Email: madhu@cs.harvard.edu. §School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts, USA. Supported in part by a Simons Investigator Award and NSF Award CCF 1715187. Email: svelusamy@g.harvard.edu.