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Revision #1 to TR21-011 | 24th February 2021 16:36

#### Classification of the streaming approximability of Boolean CSPs

Revision #1
Authors: Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy
Accepted on: 24th February 2021 16:36
Keywords:

Abstract:

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. 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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ 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 [Guruswami-Velingker-Velusamy APPROX'17], [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to explore 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.

Changes to previous version:

Improved the presentation.

### Paper:

TR21-011 | 13th February 2021 20:01

#### Classification of the streaming approximability of Boolean CSPs

TR21-011
Authors: Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy
Publication: 13th February 2021 20:11
Keywords:

Abstract:

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. 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 completely characterize the approximability of all Boolean CSPs in the streaming model. Specifically, given $f$, $\gamma$ and $\beta$ we show that either (1) the $(\gamma,\beta)$-approximation version of Max-CSP$(f)$ has a probabilistic streaming algorithm using $O(\log n)$ space, or (2) for every $\epsilon > 0$ the $(\gamma-\epsilon,\beta+\epsilon)$-approximation version of Max-CSP$(f)$ requires $\Omega(\sqrt{n})$ 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 [Guruswami-Velingker-Velusamy APPROX'17], [Chou-Golovnev-Velusamy FOCS'20] by giving a systematic way to explore 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.

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