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} → {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} with uniform marginals. Our positive results show wider applicability of bias-based algorithms used previously by [GVV17] and [CGV20] by giving a systematic way to discover 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. This paper replaces the paper [CGSV21] by the authors. The previous version had errors and is now withdrawn. 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.

[1]  Raghuvansh R. Saxena,et al.  Multi-Pass Graph Streaming Lower Bounds for Cycle Counting, MAX-CUT, Matching Size, and Other Problems , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[2]  Amit Chakrabarti.  Data Stream Algorithms Lecture Notes , 2020 .

[3]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

[4]  Elchanan Mossel,et al.  Approximation Resistant Predicates from Pairwise Independence , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[5]  Sudipto Guha,et al.  Tight Lower Bounds for Multi-pass Stream Computation Via Pass Elimination , 2008, ICALP.

[6]  Venkatesan Guruswami,et al.  Streaming Hardness of Unique Games , 2019, APPROX-RANDOM.

[7]  David P. Woodruff,et al.  On the exact space complexity of sketching and streaming small norms , 2010, SODA '10.

[8]  Andrew McGregor,et al.  Graph stream algorithms: a survey , 2014, SGMD.

[9]  T. Sanders.  Analysis of Boolean Functions , 2012, ArXiv.

[10]  Mikhail Kapralov,et al.  An optimal space lower bound for approximating MAX-CUT , 2019, STOC.

[11]  Andrei A. Bulatov,et al.  A Dichotomy Theorem for Nonuniform CSPs , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[12]  Prasad Raghavendra,et al.  Optimal algorithms and inapproximability results for every CSP? , 2008, STOC.

[13]  Venkatesan Guruswami,et al.  Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph , 2017, APPROX-RANDOM.

[14]  Ke Yi,et al.  Data Stream Algorithms , 2010 .

[15]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[16]  Sanjeev Khanna,et al.  Streaming Lower Bounds for Approximating MAX-CUT , 2014, SODA.

[17]  David P. Woodruff,et al.  New Characterizations in Turnstile Streams with Applications , 2016, CCC.

[18]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[19]  Ran Raz,et al.  Exponential Separation for One-Way Quantum Communication Complexity, with Applications to Cryptography , 2008, SIAM J. Comput..

[20]  Madhu Sudan,et al.  Classification of the streaming approximability of Boolean CSPs , 2021, Electron. Colloquium Comput. Complex..

[21]  Aaron Potechin.  On the approximation resistance of balanced linear threshold functions , 2019, STOC.

[22]  Madhur Tulsiani,et al.  A characterization of strong approximation resistance , 2013, Electron. Colloquium Comput. Complex..

[23]  Dmitriy Zhuk,et al.  A Proof of CSP Dichotomy Conjecture , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[24]  David P. Woodruff,et al.  Turnstile streaming algorithms might as well be linear sketches , 2014, STOC.

[25]  Sanjeev Khanna,et al.  (1 + Ω(1))-Αpproximation to MAX-CUT Requires Linear Space , 2017, SODA.

[26]  Optimal Streaming Approximations for all Boolean Max-2CSPs and Max-ksat , 2020, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS).

[27]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[28]  Piotr Indyk,et al.  Stable distributions, pseudorandom generators, embeddings, and data stream computation , 2006, JACM.

[29]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[30]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).