期刊名称：Electronic Colloquium on Computational Complexity
印刷版ISSN：1433-8092
出版年度：2021
卷号：21
语种：English
出版社：Universität Trier, Lehrstuhl für Theoretische Computer-Forschung
摘要：A Boolean constraint satisfaction problem (CSP), Max-CSP(f), is a maximization problem specified by a constraint f:−11k01 . 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 () -approximation version of the problem for parameters [01] , the goal is to distinguish instances where at least fraction of the constraints can be satisfied from instances where at most 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, and we show that either (1) the () -approximation version of Max-CSP(f) has a probabilistic dynamic streaming algorithm using O(logn) space, or (2) for every 0 the (−+) -approximation version of Max-CSP(f) requires (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 −11k with uniform marginals. Our positive results show wider applicability of bias-based algorithms used previously by [Guruswami-Velingker-Velusamy APPROX'17] and [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to discover biases. Our negative results combine the Fourier analytic methods of [Kapralov-Khanna-Sudan SODA'15], 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.
